Which wires should I use for stripboards?
I like solid magnet wire or wire-wrap wire of about AWG30 for soldered connections. The thin insulation does not tend to allow the wires to bend preferentially where they are stripped and solid is easier to deal with. Polyester magnet wire can be soldered through so there is definitely no nicks, 

It also pays to get a good stripper- if you are nicking the wires and a few strands are damaged, then it will break off at the insulation all the faster. A thermal stripper works well for some insulation types. For Kynar wire-wrap wire I have a little mechanical German-made doodad that works really well (not cheap though). 

Stranded wire is best handled by threading it through a hole then soldering it flat against a PCB, which is not convenient for your application (except maybe for wires going off the board to the power supply or whatever) where relatively large gauge (eg. AWG 22 or AWG 24) stranded wire is definitely the way to go.