Durham on Wikipedia.

Churches in Belmont (including Carrville and Gilesgate Moor), Framwellgate Moor, Neville's Cross, Newton Hall.
 

Baptist Church, off Sunderland Road, as seen in the background in a Streetview of 2018. NZ 2903 4280. The church website includes a history which details the places the congregation met following its formation in 1949 until the present church was opened in 1959. These include Shakespeare Hall on North Road (2018 Streetview - the green door gives access to meeting rooms upstairs, circa NZ 271 425), and a St. John’s Ambulance Brigade hut (somewhere off Churchill Avenue).

The Cathedral, dedicated to St. Cuthbert. Its full official dedication is "The Cathedral Church of Christ, Blessed Mary the Virgin and St. Cuthbert of Durham". NZ 2735 4215. © Bill Henderson. A view from an old postcard in Steve Bulman's Collection. Another postcard view (card franked 1929), from the collection of Mrs. Marion Allen. Another view, © Alan Blacklock. Further views of the cathedral (1, 2, 3, 4) all © James Murray. And two more © John Bowdler - 1, 2. Three more views - 1, 2, 3, the north doorway and its knocker (since replaced with a replica), the interior, the Galilee Chapel, blind arcading in the north transept, nave vaulting, detail of a carved pier in the north aisle of the nave, a cross showing Christ in Majesty, said to be from the vanished Neasham Priory (for which, see the Durham page). All © Christopher Skottowe (1962 and 1977). The East front, nave, Galilee Chapel and the clock, all from old postcards in © Christopher Skottowe's Collection. Link. Grade I listed.

Chapel of Hatfield College, © James Murray.

Chapel of the College of St. Chad. Interior view. Both © James Murray.

Former Chapel on Gilesgate. NZ 2854 4278. © James Murray. Janet Gimber has advised that this was originally Wesleyan Methodist, and was later known as Gilesgate Methodist Chapel.

Former Chapel of the College of St. Hild, which now serves as a library. The college amalgamated with Bede College in the 1980's, the Chapel of which now serves as the Chapel of St. Hild and St. Bede College. Another view. All © James Murray.

Durham School Chapel (private) as seen by the Streetview van in 2010. Built 1924-8, it stands behind the school (on Quarryheads Lane) at NZ 2678 4187. Link (includes interior photos).

Durham University Islamic Society Mosque is in a building behind the Dun Cow pub (2019 Streetview) on Old Elvet. NZ 2787 4235. Link.

Elvet Methodist Church. NZ 2770 4233. © Alan Blacklock. Another view, © James Murray, and another, © Bill Henderson (2016).

A Friends' Meeting House is shown on a map of 1860-1, off the south side of Claypath. A library was built circa 1900 (seen here in 2019, now in commercial use), but whether the meeting house remains at the rear, was demolished, or was incorporated into the new building, is not known.

North Road Methodist Chapel, originally the Bethel Chapel, dates from 1853. © Alan Blacklock. Thanks to James Murray for its current status.

Our Lady of Mercy and St. Godric (R.C., Pugin, 1864). The tower was added in 1909. NZ 2705 4264. © Alan Blacklock. Another view, and a view over the rooftops. Both © James Murray. The church website advises of a severe fire in 1985, forcing its closure for two years. Grade II listed.

Presbyterian Church on Laburnum Avenue, as seen by the Streetview van in 2014. Built sometime after 1898 (no building present on a map of that vintage), what seems to be the same building (at least it has the same footprint) is marked on maps of 1919 - 1966 as a synagogue, in 1967 as a Pentecostal Church, and thereafter as" Ch." or "PW". NZ 2653 4240. Link.

The former Primitive Methodist Chapel on Back Silver Street (NZ 2731 4249), as seen by the Streetview van in 2010. Its My Primitive Methodists entry dates it to 1826 (click on the census image). It also says that it was superseded by Jubilee P.M. Chapel on North Road from 1861. The Jubilee Chapel was demolished in 1961, and its site (now Iceland) can be seen here in a 2018 Streetview, though the chapel stood further back than the current frontage.

St. Anthony's Priory (Anglican) stands off Claypath at NZ 2767 4264, in what was the vicarage of St. Nicholas Church. It is probably the building in the distance immediately above the silver car seen in this 2019 Streetview. Link.

St. Cuthbert on Framwellgate Peth. NZ 2678 4312. © James Murray. Another view, © Bill Henderson (2016).

St. Cuthbert (R.C.) at Elvet. NZ 2780 4228. © James Murray. Two interior views - 1, 2, both © Mike Forbester. Link, and history here). The grade II listing says it was built in 1826-7; the tower and Lady Chapel added in 1869.

St. Giles on Gilesgate. Parts of it date back to the 12th century, when it was founded as part of St. Giles Hospital. NZ 2838 4267. © James Murray. Another view, © Alan Blacklock. Grade I listed.

According to this source there were two bridge chapels on Elvet Bridge. St. James Chapel, at the western end (NZ 2752 4243) and at the opposite end, St. Andrew's Chapel (NZ 2757 4240). Here's a view of the western end of the bridge from Streetview. Apparently a fragment of St. Andrew's survives, but the photographs I've been able to find on-line don't indicate exactly where - but here's a general view.

St. Joseph (R.C., 1966) on Belles Ville (originally Mill Lane), as seen in a 2016 Streetview. NZ 2930 4272. Link.

Some O.S. maps mark "Site of St. Leonard's Chapel" just to the north of St. Cuthbert, at NZ 2675 4318. The point marked is to the left of the large white building, as seen here on a 2016 Streetview.

St. Margaret of Antioch. NZ 2711 4239. © Alan Blacklock. Grade I listed.

St. Mary le Bow stands on North Bailey, at NZ 2746 4216. It's now a heritage centre. © Bill Henderson. Grade I listed.

The ruins of St. Mary Magdalene medieval chapel stand at NZ 283 429, immediately north of the A690. Another view. A plaque gives a little history - the chapel was also a hospital, which at the time served the spiritual as well as the physical needs of the patients. It was also a parish church. All © Martin Richter (2012). Link1. Link2. Link3 (which includes a scale drawing). Grade I listed.

St. Mary the Less, the chapel to St. John's College, is a former parish church. NZ 2733 4195. © Alan Blacklock. Another view. © James Murray. Link. Grade II listed.

St. Nicholas (1857-8). NZ 2740 4257. © Bill Henderson. Another view, © Alan Blacklock. Interior view, © James Murray. James advises that the ecclesiastical usage of the church was turned by 90 degrees in the 1990's. The site of the original high altar. Another view of the church, © John Bowdler. An attractive old postcard view of the market place and St. Nicholas. From Andrew Ross's Collection. Link. Grade II listed.

St. Nicholas' Cemetery on Providence Row has a mortuary chapel, as seen by the Streetview van in 2010. NZ 2766 4291.

St. Oswald. Another view. NZ 2762  4191. Link. Its grade II* listing says that it dates from the late 12th century, and is probably on the site of an earlier church. St. Oswald Church Institute. All © Alan Blacklock.

The site of another medieval chapel - St. Thomas the Martyr - is also indicated on some maps, at NZ 2770 4279. North of Claypath, the site is now near the western end of Hill Crest. The site indicated is behind the property seen in this 2014 Streetview.

Salvation Army on Saddler Street incorporates part of the North Gate of the castle. NZ 2747 4236. © James Murray. Grade II* listed.

The Spiritualist Church on John Street can be seen here on a 2014 Streetview. NZ 2675 4247. Link.

Former U.R.C. (1905) on Claypath was previously Congregational. NZ 2757 4267. © James Murray. Previously unnamed, thanks to David Regan for providing the identification. This source lists 3 previous chapels on the same site since 1751. Since James took his photo, the church has become Christchurch - link. Grade II listed.

Waddington Street U.R.C. (1872) was originally Presbyterian. It can be seen in Streetviews here (2019) and here (2018). NZ 2675 4265. Link, history here.

A Wesleyan Methodist Chapel is shown on an 1861 O.S. map at NZ 2770 4243, on what was then Chapel Passage. By 1919 it was being indicated as a Salvation Army Hall, and the last map showing it is of 1938-9. Since demolished (date not known), the site can be seen on a 2019 Streetview here, about where the two white cars stand, visible between the gates.

Belmont (including Carrville and Gilesgate Moor).
The site of the former Dragon Villa Wesleyan Methodist Chapel, at the S.W. corner of the Renny's Lane and Dragon Lane junction, at NS 2967 4281. It seems to have been built at the end of the 19th century, was still extant in 1940, and appears to have been demolished relatively recently. © Martin Richter (2019).
Emmanuel Church (Newfrontiers) meet in The Durham Centre on Belmont Industrial Estate. The building is in the centre of this 2009 Streetview. NZ 2984 4417. Their website history page lists several other venues following their foundation in 1980. They first met in a school building in Whinney Hill, perhaps in the demolished Whinney Hill Secondary Modern School, seen here by Streetview in 2009. Next, until 1996, they met in a former carpet factory, at NZ 2743 4273, off Freeman's Place, then in a Sixth Form Centre east of Freeman's Place, at NZ 2750 4288. I think this can be seen here in a 2020 Streetview, but please feel free to correct me if I'm wrong (as with anything else!).
The Methodist New Connexion had two chapels, both on the north side of Front Street. The first, at circa NZ 2951 4242 was of 1835, and the second a little further east at NZ 2966 4242. Unfortunately the available maps aren't precise as to the location of the first chapel (i.e. the exact building is not indicated). The site of the second chapel was where St. Francis was later erected.
The Primitive Methodists first established themselves in 1839 on the north side of the main road through Carrville. This was at NZ 3050 4395. It's today in commercial use, and can be seen on a 2015 Streetview. It was in use for only a relatively few years, and was replaced in 1868 by another chapel on the other side of the road, at NZ 3056 4402. It was replaced on the same site by the present Methodist Church in 2002. © James Murray.
Another Primitive Methodist Chapel was built in 1852 (source) at Ernest Place on Gilesgate Moor. It survives, and can be seen here in 2020. The date-stone however, which can be seen by zooming in, seems to say 1862, not 1852. Its closure date is not known, but it was before 1940. NZ 2947 4280.
St. Francis (CoE) was a re-named Mission Church. It stood at NZ 2969 4240, on Front Street. The Streetview van viewed the site when it last passed by in 2020.
St. Mary Magdalene, on Broomside Lane. Its grade II listing dates it to 1857, by William Butterfield. A 2016 Streetview provides an alternative view. NZ 3038 4360. © James Murray.
There was a Wesleyan Methodist Chapel on Bainbridge Street in 1835. Demolished at some undetermined date after 1896, the site is now part of the back garden of a house, as seen here by Streetview in 2009. NZ 3083 4401. It was succeeded by Bethel Chapel in 1881, at NZ 3069 4422. This survives in commercial use, and can be seen in Streetviews here (2019) and here (2009).
 

Framwellgate Moor.
Methodist Church (formerly Ebenezer, 1869) on Front Street. A map of 1896 labels it as Methodist New Connexion, and later it was United Methodist. NZ 2641 4449. © Bill Henderson.
A Primitive Methodist Chapel (1870) once stood on Close Row. Chapel and street have both gone - the street was replaced by North Terrace (to the north) and Garden Avenue (to the south). It stood at NZ 2637 4462, and the site can be seen here on a 2014 Streetview. It faced to the right, roughly in the area of the greenhouse and red car.
St. Aidan (1871) on Front Street. NZ 2652 4472. © Bill Henderson. Link.
A Wesleyan Methodist Chapel once stood on the west side of Front Street at NZ 2644 4464. Two shops now occupy the site which can be seen on this 2019 Streetview.
There was another Wesleyan (now demolished) in Pity Me, which stood at NZ 2650 4537, a little way west of Woodbine Road. Its site was passed by the Streetview van in 2009.

Neville's Cross.
A Methodist Church (originally Primitive) once stood on Neville's Cross Bank, at NZ 2614 4195. There is a reference on-line to it having been a tin tabernacle in 1940. The two-storey modern building on the site was seen by the Streetview van in 2018.
St. John the Evangelist (1896) as seen by the Streetview van in 2010. NZ 2630 4198. Link1; it's history page advises that St. John was a daughter church of St. Margaret at Crossgate, and was commonly known as the church in the fields. Link2.

Newton Hall
All Saints on Carr House Drive, as seen by the Streetview van in 2017. It was built in 1967, and is now home to Anglicans and Methodists, as a LEP. NZ 2741 4502. Link. The brief history included on the Methodist website mentions services having been held at one time at Blue Coat School, which stands on the east side of Langley Road at NZ 2841 4477, seen here in 2019. The wording isn't very clear, so was it the Anglicans or Methodists who worshipped at the school, and where was the Methodist Church, stated as having been built in 1967?
The Roman Catholic Church, St. Bede, was closed as unsafe in 2011 and demolished by 2014. It had only been built in 1970. NZ 2705 4494. Seen here and here in 2009 by Streetview, the congregation now meet at the chapel at St. Godric's R.C. School (2018 Streetview) nearby, which is at NZ 2712 4500. Link.