Prince played Hohner Madcat and Dave Rusan Cloud guitars. Keyboards include a Yamaha DX7, a Fairlight CMI, an Oberheim OB-8, and an Ensoniq Mirage. The main drum machines were a Linn LM-1 and the Fairlight. Guitars and the Linn were all processed through Boss effects.
You can find a full list on the Sign O The Times era page.