The best thing to do to a Mazdaspeed with regards to the driveline is to pull the 6-speed and drop in a 5-speed. You'll have better gear spacing, a taller top gear (really!), better shifting and you'll probably make some money off the deal.
FM actually offers both turbos and supers, but overall I think the turbos work better. You'll find that everyone at FM agrees, actually. Where the superchargers are fun is when you want to kick the tires free - they're the hoodlum cars. But more important than if your compressor (or blower) is driven by a turbine or a belt is the implementation. Overall, I feel there's been more effort put into the turbos and a lot more progress over the years.
Twin-screws aren't seen very often for a few reasons. Kenne Bell (IIRC) had an Autorotor-based kit that used the 0.89l compressor, and I believe they have an exclusive for that unit on the Miata. Their kit wasn't a great one, but the end result is that the 0.89 isn't available. The 1.2l has been used, but it turns out that it really only works well when it's wound up tight to at least 12 psi - and then you end up running perilously close to the redline of the compressor. They had a failure rate on the compressors that was a bit too high to be comfortable, but when they're working they're a riot to drive. Very fun, but not really available right now.
The M45 (aka Sebring aka Downing/Atlanta aka Jackson Racing) supercharger kit has been around as long as the Miata has. It was originally designed for a 6 psi application on a 1.6, and that's where it works best. Trying to extract more hits the point of diminishing returns almost immdiately and it has a few driveability quirks. Still, if you want to bolt one on and then never mess with it again, it's a decent setup.
The MP62 Eaton has become the blower of choice, and I believe GRM is going to be playing with one shortly. There are good implementations and bad implementations on the market - there's one kit in particular that shows a very high rate of engine failure, mostly due to fuel control problems and knock. The MP62 is a blower that many people enjoy.