Sheffield has one of the highest retention rates of students in the country: ie they come here to study and then stay. I was one of them - came for a postgrad for 1 year and stayed for 21.
In a previous job I spent a lot of time in other cities and I can honestly say Sheffield has one of the nicest atmospheres. Every place has social problems but like most places they tend to be geographically concentrated (sadly for the people stuck in those ares). The south and south west of the city, as others have said, are leafy and as safe as you'll find anywhere. Also very middle class, which may or may not bother you (eg depending on whether you mind the subtle pressure to 'keep up with the Joneses' and have the cash to shell out for pricey school trips).
Millhouses, Ecclesall, Carterknowle, most of Nether Edge (bit more studenty) also lovely places to be with a mix of housing (Victorian terraces and semi-detached with small gardens; 1930s semi-detached with big gardens; 60s/70s estates with HUGE amounts of indoor space by UK standards) and also 'good/outstanding' schools (depending on what matters to you about a school): High Storrs, Silverdale, King Ecgberts, .
All on the other side of the city to Ice Sheffield though. But on the right side if you're likely to be going to Nottingham.
It doesn't feel like a big city, more like a series of very well connected small towns where you'll almost always find that you have a friend in common when you meet someone new. Great cultural life too. Lots of music, arts, theatre, and the kind of place where if people want something to happen, they just get on and do it themselves.