I'd certainly suggest strobe/flash over constant lighting.
Two different ways that most people are going these days are mono light type studio lights or smaller flash units (the type that you might use on the camera's hot shoe). Both could be used on light stands with the usual modifiers like umbrellas or softboxes etc.
There are pros and cons to each. Studio lights need AC power or a big batter pack while flash units usually run on AA batteries. The flash units are a lot smaller as well. So if you are only shooting in your studio, it probably doesn't matter...but if you want to bring your lights to shoot wedding formals, then it's a lot easier to bring several flash units than several studio lights.
You may need flash units for shooting weddings anyway, so it might be worth your while to buy flashes that could do double duty.
Studio lights will usually have more power than flash units...but that probably won't matter in such small space.
Once you have that decision made, there are several options. It will help to know your budget and maybe your expectation of quality. Do you want to buy lights that are more expensive but that will probably last a lot longer than cheaper ones. Or are you content to keep your initial investment lower?