I've been to London 10 times in the past eight years or so, and seven of those trips were solo. I feel very comfortable there, even at night after a show. As others have said, stick to the more populated/tourist areas and you should be just fine. And..as you would anywhere, be aware of your surroundings and look
both ways before you cross the street !
On all of my trips I've stayed in the Kensington/South Kensington areas -- usually at either the Copthorne Tara or the Rembrandt. The Copthorne Tara is just behind the High Street Kensington tube station (Circle/District), but it's a bit of a dark walk back to the hotel after hours -- nothing to worry about, however. The Rembrandt is around the corner and a short walk from the South Kensington tube station (Piccadilly/Circle/District) but is directly across the street from the V&A Museum, and the #14 bus stops right out front.
Very convenient!
On my next trip I've decided to try the Crowne Plaza Kensington -- looks like a great neighborhood with loads of restaurants/groceries/etc., and the Gloucester Road tube station is also on the Piccadilly/Circle/District lines -- one of those three lines will get you almost anywhere you want to go in London.
Familiarize yourself with the tube map, some bus routes, and some main and side streets in the area you'll be staying/exploring and you'll be fine. Have a great time!