Obviously there are many recurring characters over the span of the Star Trek canon.
Some obvious examples: Q, Lwaxana Troi, Gowron, Lursa, B'Etor, Adm. Nechayev, Lt. Barclay, Gul Dukat
What ship appears in the most episodes/films?
What is not admitted:
- support craft for the above ships (Delta Flyer, shuttlecraft)
- space stations and docks
- ships from different eras with the same name which are not the same ship. (Do not argue that we technically don't know whether a TOS-era and a TNG-era ship are the same or not - there's no precedent for that)
- multiple mentions/appearances/instances in the same episode/film count as one
What is admitted:
- refits are the same ship
- mentions of the ship in dialog
- on-screen mentions/representations of the ship on displays
- we need positive identification. The ship must be positively identifiable. For example, in a battle or debris scene, we must be able to either see the name or serial number or it must be mentioned in dialog. No speculating that it's "probably a certain ship". The basis for that speculation would usually already count as a mention (e.g. previous dialog)
- parallel universes, hallucinations, Q-creations, etc are in-bounds as long as it represents 'the same ship form the same era'
COMMENTS ON THE GIVEN ANSWERS
It's become clear I'm not going to make anyone happy by selecting an answer.
There is a legitimate position that @Valorum's answer, the USS Leeds, is not a suitable answer because (a) he needed to use out of universe information to identify the ship and especially (b) appearing in the credit sequence doesn't qualify as a recurring role
There is a legitimate position that @Praxis's answer of the Stargazer in over a hundred episode because a model appears in Picards's ready room is not a suitable answer because that's not a recurring role either. For example, if Picard had a picture of Kirk on his wall, would we say that Kirk has a recurring role in hundreds of episodes?
Of the remaining answers, it looks like @kl78's answer of the Rotarran has the biggest role.
I actually believe @Shwern's answer may be the most useful one, but since it doesn't currently subsume @kl78's answer, it can't be correct.
I have my personal position on when I try to focus on the intent of the question, but others get upset when I clarify after they've already given answers. Therefore I will not accept an answer and I ask everyone please just acknowledge there are merits to all the answers and we'll leave it at that. Everyone please just enjoy yourselves -- I think all the answers are interesting.