I live in North Greenwich, it is just south of the river but very well connected to Central London, it is safe and some people refer to it as "Yuppie Greenwich". The cheapest one bedroom flat will fetch you £210000 ($420000) or thereabouts, mine is 2 bedroom and was bought for £265000. Given the current property price crisis I would expect price to be now similar (it would have been close to £280000 6 months ago).

Some parts in Woolwhich, South of the river, are good (the closer to the River the better), the DLR is extending there.

The East of London is in general less safe, I would avoid places like Thamesmead Sidcup and Lewisham like the plague, places around Withechapel are also difficult, places around Limehouse can be very nice but amenities are not very good. The Isle of Dogs, south of Canary Wharf, may be getting better. With all the construction going on in recent years in close proximity to Canary Wharf other businesses have sprout up under the sky scrappers shadows, but keep in mind that all the Tower Hamlets borough is considered the most deprived are in the UK!