The question is which city has more chavs? Manchester or Birmingham?
It depends if you really do mean chav or maybe scally? There is a difference.
In Liverpool ( my home town ) there are plenty of scally's but no chavs.
But to answer you're question, I'd say that Manchester has more scally's and Birmingham has more chavs.
oh, and my favorite city is Gdansk, followed by Torun.
Gdansk because of the old town/beaches/forests/lakes close by and Torun cause it's small and has a nice old town/city centre.