Coming late to this! And what a loaded one it is too.
Basically, Sheffield suffers (more than a lot of other cities I know of) from "perception syndrome." People tell you certain areas are "better", therefore people who want to move to "better" areas move there, and so on. It's terribly ghettoised.
S10 and S11 are very popular, although they cover a plethora of sins - S10 includes Broomhall, which as Hula has said is a bit "crunchy" - has some nice Victorian villas but also utterly vile bits - and Crookes, which is friendly but rather "shabby-genteel" and has a lot of student roads. Absolutely chock-full of terraces, which form the backbone of Sheffield's housing stock. So is Hunter's Bar, in S11, which is near the trendy Ecclesall Road area, and Endcliffe Park which is where all the yummy-mummies hang out before taking their darlings to coffee-shops. More expensive are Ecclesall, where Nick Clegg hangs out, and Ranmoor.
If you are a footballer or a nouveau-riche with more money than sense, you will fit in in Dore (S17).
Under-rated, and nice if you know where to look - is S6, also to the west of the city. There's Walkley - OK, cheap and studenty and decent curry places and that sort of thing; Stannington - residential, lots of newish houses, and quite nice, but be careful which school catchment you are in. Avoid Hillsborough - supposedly on the up but still a bit drab and quite chavvy at weekends. Nicest bits of S6 are Bradfield - very rural, good schools - and Loxley, rural-suburban with a good village atmosphere and again, very good schools.
(Of course although I hate the things, if you are looking at the state primary schools with the best results, you'll inevitably go to the league tables. The one my children go to came 4th this year!)
it's a very friendly city and as a southern ex-pat I heartily recommend it!