oBSG

A partial list can be constructed from the dialog in the premier of the original BSG; several more from subsequent episodes.

From the Wikipedia entry on Battlestars:

Galactica, Atlantia, Pacifica, Triton, Acropolis, Pegasus, Columbia, and Rycon are all mentioned on film.

Solaria, and Cerebus are mentioned in the novelizations.

Prometheus was on a list given to the production department.

That makes 11 mentioned in OBSG sources. We don't have a canon affiliation to given colonies, either. Adama heads for Caprica because it's home to him... to try and rescue his wife.

nBSG

The Galactica is one of the original 12, according to the miniseries.

According to the Battlestar Wiki, 3 are known from the nBSG video: Galactica, Columbia, and Athena. These belong, according to the designer (as quoted in the wiki), to the Jupiter Class, which means we know 4 of the 12.

The Pegasus is not of the Jupiter class, which probably rules it out, as the Mercury class is newer. It might, however, have been a replacement for a Jupiter class Pegasus.

Galactica,Atlantia,Pegasus,Pacifica,Caprica,Triton,Acropolis,Rycon,Solaria,Cerebus,Athena,Orion.

Galactica, Atlantia, Pacifica, Solaria, Columbia, Triton, Acropolis, Rycon, Cerebus, Athena, Orion & I think that Archeron is one

Twelve colonies/tribes are mentioned, and yet, we see only five flying in formation during the opening credits. Five Battlestars and no screening or escort vessels. Also, logically we should discount the PEGASUS as it was thought lost, while under the command of Commander Cain in some battle in the past. It was a surprise to all aboard GALACTICA including Adama, who presumably was old enough to have remembered, or even served with him. Cain did seem to remember Adama as a fellow Colonial warrior. Also, and forgive my 64 year old memory, wasn't the Colonies at war with the Cylons a thousand year's war, which ebbed and flowed over time. In any case the battlestars would have had to be part of different classes. They wouldn't just keep cranking out the same class. As new technology was made available it would have to be integrated, and over time the frames would need to be replaced. So it stands to reason they would have different classes.

This is in regards to the original series.

Eight Battlestars were seen or mentioned by name during the TV series. Of these, five were known to have survived by the time the original movie took place.

Many are identified in background radio chatter in the original pilot film ("... Battlestars Atlantia and Acropolis losing speed..") (Background voiceover Male: "...Request damage reports from Battlestars Triton and Pacifica.." Background voiceover female: "...no response from the Triton and the Pacifica") (Background voiceover female: "...This is the Acropolis. We're on fire. Request shuttles to..." )

A sixth Battlestar (the Pegasus) was thought to have been lost, but reappeared during the series.

• Battlestar Galactica

  Built 500 yahren before the close of the Thousand Yahren War. It is commanded by Commander Adama. The Galactica has about 150 Vipers aboard, a mixture of its own, some from the other Battlestars at the Peace Conference, and a great number of fighters from the Pegasus.

• Battlestar Atlantia

  Lost at the Peace Conference. Atlantia carried President Adar and the Council of Twelve. Was the first battlestar destroyed in the Battle of Cimtar.

• Battlestar Pacifica

  Lost at the Peace Conference. Novelization states that she was sister ship of the Atlantia, was the largest of the battlestars and had previously been destroyed.

• Battlestar Triton

  Lost at the Peace Conference.

• Battlestar Acropolis

  Lost at the Peace Conference.

• Battlestar Pegasus

  Sole survivor of the Battle of Molecay, 2 yahrens before the Destruction of the Colonies. The Pegasus was commanded by Commander Cain. Missing in action since the Battle of Gamoray when she destroyed two Cylon Basestars. Viper pilots from the Pegasus' Silver Spar Squadron wore a helmet design that bore a black 'flying horse', unlike the Galactica's Blue and Red Squadron pilots' falcon/eagle crest Viper helmets, due to the Pegasus' own namesake.

• Battlestar Columbia

  In the episode "Gun on Ice Planet Zero", a Cadet Cree claims to be from the Battlestar Columbia in an attempt to foil the interrogation efforts by the Gold Command Centurion Vulpa. Vulpa answers back that the Columbia was lost in the Peace Conference with the rest of the fleet.

• Battlestar Rycon

  In the episode "Take the Celestra", during an awards ceremony Commander Kronus is described as having 'risen from the ranks to command the Battlestar Rycon, a ship famed for destroying 3 Cylon Basestars at the Battle Of Cosmora Archipelago, but at the cost of the Rycon herself presumably.

• Battlestar Solaria

  Another ship mentioned as destroyed in the novelization of the original series first episode. Was the last Battlestar destroyed according to the Battlestar Galactica novelization, when Lt. Starbuck and Lt. Boomer tried to find another ship to land on because of burning up a lot of fuel due to the use of full turbos on their Vipers, during the Battle Of Cimtar. Its ship insignia and the ornamental crest on its Viper pilots' helmets would have featured a 'Burning Sun'.

• Battlestar Prometheus

  Another ship that was on a list of names given to the costume department during production of the series. The Prometheus's Viper pilot helmet ornament and ship insignia would have featured a hand holding a burning torch. Rumor has it that Prometheus would be found by Galactica during the second season.

• Battlestar Cerberus

  In the novelization for "War of the Gods", Adama recalls serving with Cain and a friend named Apollo on the battlestar Cerberus as young Viper pilots. Apollo is killed in a battle at Sagitara and Adama names his son in his honor.

_{Wikipedia, Battlestar (fictional spacecraft), Battlestar Galactica (1978–1980), List of Battlestars}