From what I've seen, very few wedding photographers use studio lighting during a wedding. The main reason is probably the logistics of it. It just takes too long to set up & tear down a studio lighting kit, not to mention the issues of hauling it around. Most wedding photographers benefit from being very mobile, which allows them to easily adapt to whatever situation they find themselves in.
I think that on-camera flash is still the primary lighting tool that many wedding photographers use. Of course, the best way to improve your lighting, is to get it off the camera, so many wedding photographers are using off camera flash (OCF). This is essentially 'studio' lighting, but the main difference is that flash units are much smaller and run on AA batteries or small external battery packs. You can fit 4 flashes with batteries in a small camera bag, or in a large camera bag with your cameras and lenses.
You can put the flash units on light stands or you can use a voice activated light stand (assistant). But if you get creative, you can put them anywhere and mount them to just about anything.
Of course, if you want to bring along a few light stands and modifiers like umbrellas or softboxes, that is more to carry and more to set up...so you loose some of the portability of just flashes, but it's still smaller than studio units. But the main difference is that studio lights run on AC power. So either you need somewhere to plug them in, or you carry around a battery pack that has an inverter.
Another factor is your shooting style and whether or not you'll need the power of studio strobes or not. For example, if you like to shoot into the sun, or if you like to really dial down the ambient exposure and use a lot of flash exposure, then you may need the power of a strobe. But if not, then you're probably OK with just a good flash unit (SB 600, SB900).
Another issue is the CLS system that is built into your camera & flash. This can allow you to use your Speedlights off-camera but still retain the auto flash metering etc. You won't be able to do that with studio strobes.