Norton Rotary (in various forms) : in low-level production for quite a long time and had some racing success. Killed off mostly by business shenannigans that sunk the tiny rump Norton company. Enthusiatic owners' club (I know one).
Suzuki RE5: small production run in 1970s. Over-complication to address rotary problems made it heavy and expensive compared to equivalent conventional bike so it didn't sell well.
DKW Hercules: late '60s, early 1970s? (Don't know much about this one) Basically okay, but expensive for it's performance level so didn't sell well.
The basic problem with all Wankel engines in rotor-tip lubrication. In all the examples above, this was via a total-loss oil system using a special oil, and it's only thanks to the continued existance of the Mazda rotary cars that this oil (Shell Rotella) is still available. A bike can't carry a big oil tank, so rotary owners either have to carry extra or plan their routes around places where they know it's available. Weekend tours are fine, but world tours are out without elaborate back-up.
There was a brief fashion for turbo-charged bikes in the early 1980s. As Hobbes says, the blistering performance of conventional bikes made it's complication and problems redundant.
Honda CX500 and CX650 turbo
Yamaha XS650 turbo
Kawasaki GPZ750 turbo
There was also a Suzuki but it was never imported into the UK ( "XN85" or something like that?)
Of those, only the Kawasaki was moderately successful, and then only (mostly) because the next size up conventional bike, the GPZ900R was actually 904cc and so fell just the wrong side of the simplistic 900cc capacity limits which insurance companies used to rate bikes in those days (there was no "group" system like there was for cars: that's changed now). Speaking of insurance, note that all those bikes had medium-sized engines and so theoretically qualified for lower insurance than bigger conventional ones. This was the probably the primary reason for their existence, but the insurance companies quickly wised up and stuck a premium on anything with a turbo, so even that rationale quickly evaporated.
A fundamental problem which affects all turbo bikes is throttle lag. Basically, you open the throttle and get some more power instantly, but then the power keeps building for a constant throttle setting as the turbo winds up to the new speed. This is very disconcerting: you open the throttle for the next straight and the bike speeds up, then just before the next bend it speeds up again all on it's own! The basic solution in cars is to put the turbo right next to the exhaust ports, thus minimising the delay before the exhaust gas velocity at the turbo increases, but it's hard to package a bike like that, and three of the above had the turbo behind the engine with long pipes leading to it and consequent turbo-lag. The Kawasaki was better than the rest because it's turbo was in the belly pan on the end of relatively short exhaust headers, but that in turn brought it's own problems with water, rust and dirt.
BMW were great fans of supercharging back in the 1950s, the layout of their flat twin engines being particularly suitable. They keep looking at it again every few years, but no production hardware has ever resulted. Peugeot, however, have produced a small-capacity supercharged scooter which has been relatively successful.