Tallinn seems to be a quite save destination for travellers. You need common sense like every else in the world; and like in many other places of our world there are sick people.
West of Toompea Hill in Toompark there is an old, declining sports field. This seems to be a favourite place for the local junkies to satisfy their addiction. Be prepared for some shocking scenes! (especially if you travel with your kids!)
Like any city there are places you definately dont want to be at night! Avoid Kopli area at all costs even during day time,it has to be the poorest area in tallinn so anything can happen,Also i,ve heard of instances of muggings around the ferry port area in the early hours,mainly due to "easy finnish pickings" but never the less no-one asks where you,re from when they want something!Try to avoid parks or open spaces at night,just common sense really
As in every bigger city so in Tallinn there are suburbs a tourist should avoid - firstly for your own safekeep and secondly hardly you can find any entertaining tourist attraction thereby. I would advise to keep away from the peninsula of Kopli in the region Põhja-Tallinn (you'll find a city plan at http://www.tourism.tallinn.ee/map.php to see what I'm talking about). Heroin-junkies and pickpockets over there, the rest of the city doesn't have a big problem with them. Also, Lasnamäe and Õismäe are just boring suburb-areas, but not of any danger, at least day-time.