bottom, posterior, rear, behind, backside, buttocks: these are benign and acceptable in polite, formal company.
bum, arse: these are chiefly British English, but they are understood and sometimes used by Americans.
ass: among all these words, this is probably the only "vulgar" word, but even as a vulgar word, it is mild enough to be allowable on prime-time TV. My standard for whether a word is vulgar or not is asking myself "Would I say it in front of my grandmother?" The answer for "ass" is no. Still, I use it in front of friends and other people I know who aren't offended by mild vulgarities. I wouldn't use it at a business meeting with new clients.
butt: this is about as ordinary as you can get in American English slang for this body part, and it is understood by Brits. I use it casually among friends, relatives, teachers, learners of English, and even bosses. I might avoid it if I were talking to someone who held sway over my future, like a job interviewer, judge, or parole officer.
caboose: rather old-fashioned slang. Very benign. Something an old lady might say.
fanny: Old-fashioned and benign. Something an old lady might say. I avoid it because of the confusion it may cause when speakers of British English hear it. In the UK, it means "vagina".
booty: very popular 70s slang, especially among African-Americans. It has seen a recent revival, with the additional connotation of "sex".
badonkadonk: more modern slang, chiefly African-American, for a big, curvaceous bottom. Somewhat crude: I wouldn't use it in polite company.
tush, keister: these are benign slang term that have a Yiddish feel to them. Don't forget "tuchus", which is very Yiddish!
duff: benign, but I only hear it in set phrases like "Get off your duff."
hiney/heinie: benign and humorous. Kind of cute-sounding.
patootie: old-fashioned slang. Benign and humorous, with a feeling of the American South. Often preceded by "sweet". I can imagine a Southern belle saying "You bet your sweet patootie!"
Never heard "bums", "chuff" or "prat", though "pratfall" means to fall on one's butt.
Following a habit that I picked up from Frank Sinatra, I sometimes address people as "baby", both men and women. Example: During a concert with Grace Kelly at the 'Royal Festival Hall', Sinatra said to the orchestra conductor: "Apres vous. That means 'go, baby'." Would 'baby' be OK with you?
