Mine basically, (shortened version) went like this

Guy: Do you want to come over for dinner?
Girl: *softly* No...*louder* I mean yes (followed by brief confusion)
Then the series got cancelled so we didn't actually see the date.

Well, Brittany asked Santana to come out with her but that's not really the same as going out...

^ Nope, not exactly the same.

Ermm well basically, He confessed his love for her, but she rejected him, then about a year and a half later she confessed her love for him but he rejected her, but a few months after that they are dating and going to have a baby even though you didn't really see them do owt, she just she told him she was pregnant and he was the father and they both smiled <3

Well, technically he didn't ask her out. But he did ask her to the movies, but she declined so she could practice air guitar, because he convinced her too, but of course, she wouldn't want to tell him that. xD

My current OTP is Booth/Brennan. They haven't gone on a first date. They kinda skipped over first date.

Some of my other favorites:
Spuffy- never went on a date, but were in a relationship
Doctor/Rose- Their first date was the End of the World, they called it their "first date" but they never really were together or went on a date.
Hodgela- Yeah, it was a bit awkward.

When Spike asked Buffy out in Crush it was the most awkward thing ever!

Buffy: What is this? The late-night stakeout, the bogus suspects, the flask! Is this a date?
Spike: A d— please! A date! You are completely off your bird! I mean— (he changes tone) Do you want it to be?
Buffy (disgusted): Oh my God!
Spike: You can't deny it. There's something between us.
Buffy: Loathing. Disgust!
Spike: Something's happening to me. I can't stop thinking about you. And if that means turning my back on the whole evil—
Buffy: You don't know what you mean. You don't know what feelings are.
Spike: I damn well do! I lie awake every night—
Buffy: You sleep during the day!
Spike: Yeah but— you are missing the point. This is real here. I lov—
Buffy: Don't! Don't say it. I'm going.

Sadly. ):

Sawyer/Juliet - we didn't see it
Ben/Leslie - they kissed. though Leslie kind of asked him out and he said no but for complicated reasons
Sydney/Vaughn - Vaughn asked and I don't know if I would say it was weird
Monica/Chandler - they just kissed

Well, I don't think it was awkward for either one. xP

He was about to asked out before but she was going out with his brother, so he waited. One day they were flirting and the girl said: "You know, I read a book last night...It's about a this girl and this guy and they've got this weird vibe thing going on but theres all this drama cause the girl's got this ex-boyfriend who turns out to be none other than the guy's brother. So the whole thing comes to this great big climax at this big party at their evil Dad's house"
Lucas: "Really and what happens in the end?"
Peyton: "You're just gonna have to show up to find out.""

He wasn't going but went to see her. I don't know if this count as asking him out but it's the first that comes to mind.