This song has been on TV. It was used in an Angel episode but it did cut away before the chorus. I think it was in Season 5’s “Destiny.”
Rio, by Duran Duran.
What?
OK, how about Turning Japanese, by The Vapors.
The one sung to the Colonel Bogey March. You know the lyrics:
Well, why else would they whistle that specific tune in The Bridge on the River Kwai?
“Outside Now” or “Bobby Brown Goes Down” by Frank Zappa
“Carbona, Not Glue” or “53rd and 3rd” by The Ramones
“Delia’s Gone” or “Sam Hall” by Johnny Cash
mm
Or a Rolling Stones song that was never formally released, the infamous * Cocksucker Blues*.
“We’re not a punk-rock band, we’re a new wave band…”
Mind you, some Lard {Jello Biafra vs. Ministry} would go down equally poorly: “Can God Fill Teeth?”
“Don’t ask me why, but I was walking down a street in Fairfax, California when I saw a flyer hangin’ on a telephone pole and it said “Can God Fill Teeth?” That’s right! See silver fillings turn to GOLD, and other super-normal dental happenings. Bring a flashlight and a mirror to OBSERVE!” {Cue heavy industrial noise with sampled dentist drill noises and screams}.
“War Pimp Renaissance” would probably bring the house down, too.
The Horst Wessel Lied.
pretty much anything every written by the band Cannibal Corpse…
I especially like the song titled ‘I cum blood’
Lovely stuff… 
Great, thanks for putting this into my head for the rest of the day. Damn that song is catchy…and completely inappropriate for a straight male to get caught singing in the middle of a work day. 
I’ll add “Rehab” by Amy Winehouse to the list.
any number of punk songs but let’s go for Code Blue by TSOL Alex for $200
Keeping with the Smiths, while they might “allow” it, but I would love to hear someone sing This Charming Man
Also, “Handsome Devil”
The theme song from that cartoon that used to be on Captain Kangaroo.
“Well you know my name is Simon…”
W.A.S.P. “Animal (Fuck Like a Beast)” One of my personal favorites.
It’s not edgy at all, but I sometimes wonder if a contestant asked to sing “Ten Rounds With Jose Cuervo,” would s/he be permitted to perform it? I think Bucky could have pulled it off.
Either Po Lazarus or The East is Red.
Cousin Kevin is a bully, but Wicked Uncle Ernie is the molester in “Fiddle About”: “Down with the bed clothes/up with the nightshirt…”
Yeah, but with their format the singer would only get through two, maybe three rounds.
I nom “Bitch” by Meredith Brooks(?).
Nothing by Tom Waits, clean or dirty, will ever be on that show.
“Martha” could make it. It was covered by Meat Loaf who has been on AI before.
I don’t think these obvious non-choices have been mentioned:
“Excitable Boy”, “Lawyers, Guns, and Money” - Warren Zevon
“I don’t like Mondays”
“Why don’t we get drunk and screw”
“Why don’t we do it in the road”
“I want my MTV”