Well let's compare all five franchises in likability. Keep in mind this just my opinion and reflects personality over competence. Some fields will repeat people for example T'Pol was first officer and science for example and some field will be omitted as DS9 didn't have a pilot for example. So here we go.
Captain: Sisko, Kirk, Archer, Janeway, Picard
Don't get me wrong, Picard's my favorite Trek captain, but let's face it... he's stiff and cold to outsiders quite often. Janeway just seems hard to have a conversation with at all since she's always having to overcompensate because she's the female captain as she's written. Archer's a relaxed person and someone you could sit down and have a beer with, but he also has a number of annoying characteristics. Kirk seems to be easy to get along with so long as you're a human anyways. Sometimes other races seem to rub him the wrong way. Sisko is really the guy you could just hang out with. He's the captain across the franchise that you see doing more every day things be it playing baseball, cooking and what not. Without a uniform he'd just be one of the guys.
First Officer: Kira, Spock, Riker, Chakotay, T'Pol
This one was... hard. T'Pol's just insulting and rather two dimesional, her only saving grace being she hot and wearing a catsuit. Chakotay, while not actively likable at least isn't annoying until he pulls out a script with Indian cliches in it. Riker.. ugh, he's a jerk, but at least he's a charismatic jerk. Spock, sure he's bland at times, but at least he doesn't actively go out of this way to offend outsiders. So... more by virtue of a weak field than anything else, Kira gets the win. She's genuine, will call it as it is, and despite her hard exterior really does have a softer side.
Engineer: Scotty, O'brien, Trip, LaForge, Torres
So Torres is just a sterotypical angry Klingon for the most part with little depth. LaForge is a nice guy, but just socially awkward. Trip seems to be an approachable person so long as you're human, O'brien is definitely the guy next door, but Scotty takes the cake. If anyone in Trek knows how to have a good time it's him.
Tactical/Security: Reed, Chekov, Tuvok, Worf, Odo
Odo.. I like the guy, but let's face it. He's kind of a fascist in the way he acts and by his own admission is the outsider and usually looking down on us humanoids. Worf is just the Klingon glacier who will show up to a social function because he has to. Tuvok at least is polite even if unenthusiastic. Chekov is the guy who's a little too enthusiastic at the party, but he's likable. So Reed... yes the bumbling guy who blows stuff up, takes this one for a bit of balance and a bit of a weak field.
Science/Ops: Dax, Data, Spock, Kim, T'Pol
Merged these two since they mostly are the same for storytelling purposes. T'Pol's still a Vulcan bitch, Kim's whiney and boring, Spock at least would keep quiet during a party, Data is programmed to explore humanity so he'd be a blast to hang out with, but Dax takes the cake on this one. A chick who plays tongo and drinks bloodwine? Heck yeah.
Helm: Paris, Sulu, Mayweather, Wesley
DS9 didn't have a pilot, so.. not much I can do there. Wesley's just annoying. Mayweather is slightly less so. Sulu's a well rounded person, but Tom Paris easily takes this one. Be it drinking, fixing cars, talking history or wacky salamander sex, whatever you're into chances are he is too.
Doctor: Bones, Crusher, Phlox, EMH, Bashir
This one was hard for me. Bashir has to bring up the rear because he's either annoyingly chatty or just a know it all freak, never really striking a balance between the two. The EMH, well let's face it, he -is- rude most of the time and absorbed with himself. Phlox is genuinely friendly and his naive outlook on humanity can be funny. Crusher seems well rounded and graceful wherever she ends up. But McCoy... sure he's rude, but he does it with style!
Extras: Quark, Guinan, Jake, Seven, Neelix
Well this kind of deserve it's own category. Though TOS/ENT didn't have one really and DS9/VOY had two. Anyways, this is non-scientific so whatever. Neelix is just a selfish jerk... I'm sure you've all seen my rants on the VOY forum. Seven is a hot, but... itchy with a b. Jake, well as he got older he'd be fun to hang out with. He wasn't a hero, he was just around so he's definitely one of the guys. Guinan, hey I just have a soft spot for world weary/wise bartenders. But Quark takes the cake on this one easy. This guy can have a good time and makes no qualms about it.
So... in conclusion...
TOS took 2
TNG took none
DS9 took 4
VOY took 1
ENT took 1
Not quite scientific but for the most part it sums up my feelings about the franchises. TOS had some great characters, but overall they were a bit over the top at times. TNG had probably the best chemestry in the franchise among themselves, but really were hostile towards outsiders. DS9 was literally the group of people you worked with almost, Voyager had some standout characters, but more terrible ones than good. The same with Enterprise.
So take all that as you want!