My 14 year old is generally lovely, 11 year old a little more challenging 
The trick is to ignore, as Limara says. I particularly ignore unsuitable clothing, so I have to say, I wouldn't have put your DD in a bad mood by arguing over the crop top in the first place. If DD1 wants to go out in the freezing cold with unsuitable clothing then that's her problem. I'm not the one having to walk around with skimpy shorts and no coat (but a scarf) 
Likewise, ignore slagging off to mates. I think my 14 year old is lovely, but she does slag me off on facebook, I notice (so maybe she's as horrible as any other teen!). I completely ignore this. It means nothing.
I keep the lines of communication open when the girls are civil and mostly ignore them when they aren't. If they overstep the line (by hitting each other or swearing) I just say in a very calm, cold voice, "You know that is unacceptable behaviour and I won't tolerate that again." It seems to work - probably because they know what side their bread is buttered on. They get treated to stuff - lots of dance and skating lessons which they love and a monthly allowance. I try to be as reasonable as possible when they want to do things with friends. I don't say no for the sake of it, only when I have genuine worries about their safety. They know this and so I think they generally try to keep in with me!
I do see lots of people with teens getting into what I think are pointless arguments that have more to do with control than anything else (tidy rooms, clothes, make up, homework, eye rolling, grunting, general attitude). Let them get on with those things and take the consequences for themselves outside the home. This will reduce the household arguments by masses - and they will know when you do get cross, it's for a really good reason.