Kalamaja, Kopli, Mustamäe, Õismäe, Lasnamäe

Mustamäe is really not a place with crime, because it's one of the oldest Soviet-era "Hills", and has 'grown up', so to speak. It's got lots of trees everywhere, a forest between the hospital and the university complexes, and patches of forest elsewhere, plenty of malls, and one big new cinema at Mustamäe Keskus. The district is also really compact. But since it's a sleeping area, it's got no nightlife.

Lasnamäe and Kopli have historically been more infamous, though I can't tell what it's like now. I'd venture to say, that Lasnamäe can be reasonably safe, as long as one doesn't associate oneself with questionable company.

One historically dangerous place has been Kopli Liinid on the Kopli peninsula, after which the outlying Kopli district is named. These 'Liinid are a complex of streets (liinid=lines) on one (probably eastern, IIRC) side of the peninsula with condemned and/or uninhabited wooden houses built a hundred years ago that often fall prey to arson. There are ruins. The peninsula is the endpoint of #1 and #2 tram lines.

The Kalamaja (sub)district is adjacent to Kopli near the Balti Jaam train station, and has seen gentrification. Kopli itself might one day become gentrified, too.

Õismäe and Lasnamäe are the ones with less trees. Lasnamäe is the newest and largest, and in the Soviet times, it was the endpoint for the many Russian workers and their families that came to work in the Soviet factories that were stationed here.