I really need a "Pan-Trek Kitchen Sink" to make this fit, but I'll post it in the DS9 sink since this began as an outgrowth of a response to the Bashir/Dax relationship , and also because there's an Odo/Kira conversation going on as well.

Relationships and near-realtionships that have lasted more than one episode:

Miles/Keiko
Picard/Varsh
Picard/Crusher
Riker/Troi
Sisco/Yates
Worf/Troi
Worf/Dax
Worf/Khaalar (sp? Alexander's mother)
Bashir/Ezri Dax
Kira/Odo
Kira/Shaakar
Kira/Bareil
Nurse Chapel/Spock
Neelix/Kes
Paris/Torres
Paris/Kes
Chakotay/Seska
Holodoc/Seven of Nine

Am I forgetting any?

The ones I find most convincing is probably Keiko and Miles. Sisco and Cassidy is fairly convincing, and the Neelix/Kes relationship wasn't too bad. (Neelix is a pathetically jealous lover, but that's another story.) It worked because they introduced them as a couple. If that came later, I doubt the writers would have handled it well. Riker and Troi came off as convincing, but that might well be because when we were first introduced to it, it was already past tense.

Picard/Varsh: so-so. At least it didn't intrude too much.

Kira's and Shaakar's relationship didn't seem natural; it was simply a foil for Odo, but as such it worked well. Kira and Bariel was somewhere in between. We didn't need it, but no harm was done by it. Kira and Odo: too much "tell" and not enough "show." I liked it better when Odo kept his feelings to himself and Kira had no idea.

Worf and Troi was fairly convincing, especially since it was introduced first in an alternate universe and then in a dream, but it never developed. (It was terrible that Troi never came to DS9 while Worf was with Dax. I would have loved to see Worf's discomfort at her coming, and then his confused anxiety when Dax and Troi got along so well.)

Worf and Dax were a natural couple, but I was never convinced by its execution. I would have preferred seeing a relationship similar to that of Worf and Yar: a restrained respect. (Of course, if Yar had stayed on, something else might have developed.)

Worf and Khaalar: good because of its instability; at first, Worf wanted to take the vows but she didn't, and next time, she did and Worf didn't want to. Its brevity might have contributed to its believability.

Bashir and Ezri: the writers tell us it's there, but they don't show it to us. How COULD we be convinced by it?

Chakotay and Seska: I have to go back and watch some early episodes to comment on it. For the most part, I was convinced, although it didn't seem very Cardassian of her.

Paris/Torres. Not convinced at all.

Near relationships: The Paris and Kes near-relationship was a good foil for Neelix. Unfortunately, it showed what a jerk he was. Holodoc and Seven: I still don't see how a hologram is capable of love. I just hope this remains a one-sided love. Chapel/Spock: now this was good!

There are a few more near-relationships, like Janeway/Chakotay, Kirk/Rand and Torres/Chakotay. Since the writers forgot about these (or decided not to pursue them), I will as well.

Basically, I find the relationships more convincing when it's between a regular and a non-regular (come semi-regular). Trek writers are better at handling one-sided love affairs (Chapel/Spock, Odo/Kira) than two-way lovey-dubby relationships. Personally, I wish the writers would stop pandering to the teenage soap-opera crowd by pairing off everybody. There just isn't world-enough-and-time in Trek shows to develop such parings into believable, dynamic relationships, at least not at the expense of developing what Trek shows are all about.

Forgot to comment on the Picard/Crusher relationship, and with good reason; it's pretty forgetable. It was handeled miserably in the first season, but resurrected capably in the final season. The fact that it never really blossommed is something we can all be thankful for.

While I'm thinking of it, there have also been also been hints of a Dukat/Kira relationship (all the hints being from Dukat, of course), which would never develop in a million years.

And I also forgot Garak and Ziyal. (God, there's no end to this list!) This relationship was interesting, but Garak's relationship with anybody (romantic and non-romantic alike) is always interesting.

More real relationships

Data/Yar
Leeta/Rom
Q/Q
Kirk/(women)ⁿ

Non-Romance Relations.
These are relationships that are more than just friends/co-workers, but not the like ones above.
In other words, a relationship that makes the character who he/she is and the loss of one of them would affect how we think of the other.

Odo/Quark-Hmmph, that man loves me.

Bashir/O'Brian-The loss of O'Brian from DS9 will lead Bashir to severe depression.

Kira/Dukat-Dukat takes a perverted delight in tormenting Kira.

Kai_Winn/Sisko-(a I detest Clinton or Bush feeling-yet I respect him because he's President.)

Quark/Rom-Brotherly love, Not wanting to admit that they care for each other.

Bashir/Garak-(Early on)-> Bashir was like a naive boy who gets to listen to a war veteran tell the horrors of war_but in the Abe Simpson way.

Weyoun/Odo-one like emmisary-Sisko(early DS9)

I loved Kira/Odo and Worf/Jadzia. They were the most believable and naturally developed