The whole Meydan Park which is just beside the municipal hal (Belidiye) is bustling with activity at night. People hangout here at the so many tables and chairs positioned around the fountain and the statue of Ataturk.
You can order tea, coffee and cold beverages, ice cream, sandwiches, etc at the several cafes surrounding the park.
When we arrived almost midnight, there are still a lot of people, I think even after midnight.
Since our hotel (Nur) is behind the Belediye so it's easy to get to the park and safer to walk at night on a very short distance.
Trabzon is 'lively' at night...it is a port city, so what do you expect?!! However parts of Trabzon are extremely seedy, and not really recommended at night. The teahouses in the centre of Ataturk Alani are good places for more innocent nightlife, but the best place is down at the seafront, where there are some great cafes, and in the early evening it is busy with families, couples, and oddballs such as me parading up and down the seafront promenade, eating ice-cream, popcorn, nuts, etc... Later at night, though, a local student told me it is quite a dangerous place to be. The waterfront cafes stay open every night until around 10pm, later on Thursdays, Fridays and Saturdays, and the Ataturk square teahouses seem to stay open all night. The port area around the Russian Bazaar is a dodgy place to be at night, but in the early evening it was fine.