More or less what makes wobbly bass is an automated LFO which is tempo-synced. This far, envelopes, filters and LFOs are integral part of most synthesizers. If you know how things work you can achieve similar results with almost any synth, be it a bassline, pad, or percussive sound you want to create. (When it comes to creating a particular timbre, things may get more complicated with FM-synthesis though.)
Due to possibility of waveshaping, sequencing, or otherwise modifying LFOs, Native Instruments Massive is particularly popular among dubstep producers. If you like, you can read "particularly popular" as "each and every producer is using it," which is a decent reason to try something different - deadmau5 at all uses hardware synthesizers heavily. However, it requires more than basic skill and knowledge and is more expensive. For that reason software ones dominate in the community. And you're right, on YouTube there is lots of tutorials, for almost any (widespread) synth imaginable.