Justified Trope: Bo doesn't really specialize in comedy as he does more in satire, which allows him to specialize in making fun of things, and that's why his songs can get very offensive. Plus it's funny.
Mean Character, Nice Actor: A bit complicated. Bo's arrogant, unkind stage persona is partially for effect, but every now and then his reactions with audience members blur the line. He gets pretty nasty with his audience (not just hecklers and shouting idiots) on the "Words, Words, Words" DVD, such as singing "You're not a fucking metronome" when the audience starts clapping along to the beat of a song. At the very least, he's quick to shut them down. Reports vary about how nice he actually is in person, but some of it is just the persona.
If you really think about it, the way he treats his audience is actually all part of the joke. If you really watch closely you can tell what's all just satire and sarcasm and his real vision of things becomes clear.
People who know him personally will say that the persona started as an act and that it's now more a case of Becoming the Mask.
Bo addresses criticisms like these in the finale of "what." Bo is presented with three pre-recorded speeches from caricatures that he finds disrespectful (one being someone who knew him in high school and wants to leech off his fame, one an agent who wants to meddle with Bo's act, and one just some guy who thinks he's an asshole) and Bo simply modifies their message into "we think we know you."
Additionally, compare the way he interacts with the audience in Words, Words, Words and what. When someone in the audience is being particularly loud in Words, Words, Words, he snaps at them. In what. he lampshades what they're cheering for ("Places!"), then laughs and continues the show. If anything, he's become nicer.
No Ending: Multiple times, most notably "Welcome to Youtube", but to a lesser extent "Men and Women", "The Perfect Woman", and "Ironic". Also happens in the live versions of "Oh Bo" (partly justified because the bit about the oboe is hard to replicate live) and "Words Words Words", which just kinda cut off where the studio versions have little codas and fade out.
On the DVD of "Words, Words, Words", he performs "I'm Bo Yo" as an encore, and abruptly stops in the middle of the last chorus after changing one of the lyrics to "This song's gonna end really awkwardly."
"Bo wants to make you feel comfortable. (Random voice!)"
Self-Deprecation: Often, most notably in recent performances of "I'm Bo Yo"; in the final line of the chorus he says he "can really find your G-spot", which is followed by a brief self-indicting remark.
"Oh, but I'm inadequate / Have I gotten that point across yet?"
"Art is Dead" is chock full of this.
Shoot the Shaggy Dog: In what. Bo tells a children's story about frogs, one of which experiences love at first sight and pursues the female frog, only to be eaten alive by an alligator along the way.
Stealth Pun: "Because if I'm in the closet, then you are below me / taking the B A T outta "BASEMENT", homie". Unfortunately he ruins the stealth aspect in the live version of the song available on his self-titled album.
His songs are chock full of them, in fact. "I'm Bo, Yo" is a good example.
This is essentially his modus operandi. Basically, if you laugh at tons of stuff the first time you hear the song and don't bother listening again, you're doing it wrong.