The thing that works most against my suspension of disbelief in space opera movies is that gravity is the one thing that never fails. I don’t need a real hard SF premise - hyperspace travel and really powerful realspace drives are OK, or the action would all be in our small backyard, which e.g. 2001 did well), but artificial gravity, especially artificial gravity that does not fail when everything else does, is just too convenient IMO.
Any recommendations?
I can’t think of any offhand, though something old is nagging at me. I think that Apollo 13 (not really space opera) did it fairly well for historical verisimilitude.
The problem is that for any productions filmed here on terra firma, even simulating null-G is really an expensive ‘stunt.’ There have been a few shows that have done it as a novelty, (Enterprise had a fun teaser in the first season with the gravity going on the fritz in the captain’s shower, with water drops bouncing slowly up and all,) but no matter how ‘unbelievable’ it may be, it’s just not worth the effort to do an extended space sequence that way in any futuristic setting where you could conceivably make up artificial G.
Many books, without those practical constraints, have done it well, and possibly a sci-fi radio drama could use complaints about getting around in null-G as a running gag. 
Either Red Planet or Mission to Mars (can’t remember which is which, both were 2000 flicks with similar premises) was realistic in that regards : the artificial gravity aboard the spaceship was created by spinning the habitat around a fixed axis. When the engines powering the rotation fails, so does the artificial gravity.
Was going to say 2001 actually, which had artificial gravity but also has some weightless scenes. I feel your pain, that always annoyed me about Star Trek, the ship could be completely powerless, but still have gravity (of course, can’t really blame them, I’m sure any sort of weightlessness effect that was within their budget would be distractingly bad).
Actually, now that I think about it, the last Star Trek movie with the original cast had a pretty sweet scene where a damaged Klingon ship looses gravity.
But as far as a show where a spaceship has consistent weightlessness, I got nothing. Maybe someone who watches more anime then I do knows of one. Seems like that would be a natural place to look, anyways, since you don’t really need expensive special effects to draw people weightless.
ETA: Another one that hardly counts: Contact’s one scene in the Mir space-station is weightless. Course the Mir is (or was) a real space station, so they could hardly do it otherwise.
In Babylon 5, the Earthers didn’t have artificial gravity technology, and so simulated it by spinning the station. And really, there’s no way that kind of artificial gravity can fail, without the entire station being destroyed. They did have some more advanced races with artificial gravity, though (which never seemed to fail no matter how badly the ship was damaged), and there were a few continuity glitches where people are seen standing and walking around on ships that shouldn’t have had artificial gravity.
Or without the piece of the station that you’re in coming loose and shooting out on an inertial straight-line course. If the gravity suddenly goes away on a spinning space station, you should know that’s what’s happened to you.
What you do about it is a different story. 
Stuff without artificial gravity:
**2001
2010
Destination Moon
Operation Moonbase
Man Into Space (TV series)
Conquest of Space
**
I note that two of those are based an Arthur C. Clarke novels (you can quibble about 2001, but it’ds ultimately based on The Sentinel), and two on Heinlein works. And Jerome Bixby wrote for Man Into Space. SF writers liked accuracy. TV and movies didn’t because, to quote from Heinlein’s own essay on the making of Destination Moon, "the answer to almost every question in Hollywood that starts with ‘Why don’t they…?’ is ‘Money.’ " Zero G is expensive, as Heinlein’s essay and Jerome Agel’s book The Making of 2001 will easily show.
The British serious Star Cops (much more serious than its title suggests) at least acknowledged the problem, even if they couldn’t address is consistently with their low budgets.
Gah. That’s even dumber than just ignoring the issue to begin with.
Yeah, if the rotation mechanism fails, then what you’ll end up with is the entire ship rotating more slowly. And that’s assuming that there’s a significant portion of the ship that isn’t rotating, when you needn’t have any such part.
Courtesy of David Szondy’s Tales of Future Past:
There’s the old BBC series Moonbase Three. If that counts; the Moon has gravity, after all.
And Conquest of Space, which was really more of a dramatized documentary.
And Moon Zero Two.
Hijack: Y’know what I’d like to see? A basic-cable network that would be a fusion of the Sci-Fi Channel, TVLand and Turner Classic Movies. I.e., nothing but old SF movies and TV shows. There’s plenty of material out there – much of it never seen by most contemporary viewers – to fill out the hours.
That’s the one I was going to mention. I don’t remember any inconsistencies with zero-G. Much of it took place on the moon, and I never saw any attempt to portray less-than-one-G.
Moonraker
I mentioned it above. George Pal made it, so I sort of think of it as fallout from Destination Moon (and Pal was already set up for zero-G effects). But DM was more of a documentary – Conquest of Space gave us a religious-nut commander who almost sabotaged his own mission to Mars.
In The Black Hole while the Cygnus had AG the Palomino did not.
I’ll second Babylon 5, at least in part.
I believe the Centauri have artificial gravity, but the Humans/Earthers don’t. I
Did the Minbari have artifical gravity?
Yes, but the Narns do not.
Also Earth2, the original if anybody remembers that.
When ZeroG Corporation launched a few years ago the hope was that people would be lined up to utilize them, to film movies, TV shows, commercials, music videos, whatever. Except for a very few cases, that has not happened. Mainly because of the cost, but also because it’s just not very practical. It is advertised as “up to 30 seconds” of zero g at at time, but it’s closer to 20. And it is not consistent. People float up and hit the ceiling, then drop to the floor, then float towards the front of the plane, then to the back. Would be difficult to piece together a realistic looking scene when you’re only getting a few seconds of usable footage per parabola.
The ship in Red planet had two habitats spinning in opposite directions to stop the rest of the ship from rotating. An engine failure as depicted in the film would result in weightlessness.
Oh, and don’t forget the greatest movie ever made, Barbarella.
There was a movie called Thieves and Prostitutes that, legend has it, was approved by the Russians to be filmed aboard Mir, and it was to include a love scene. An actor and actress went through astronaut training, but then Mir came down and that was the end of that. They went ahead and made it anyway using special effects instead. It was actually pretty well done.