Which type of veneers requires no or very minor tooth preparation?

Whatever it takes... In the past the thought was to do minimal preparation as if you wanted to return to what your teeth looked like before..A reversible procedure? The reality is a change for the better is what you want, so go forward , make the change, allow whatever preparation is necessary to end up with natural looking teeth, the right size & aesthetically pleasing that do not irritate the gum neighbors. Do it.
Lumineers. You may have heard about a no- prep procedure called lumineers (not the music group) where little (if any) preparation is done to the teeth prior to veneer placement. Keep in mind though that the newest techniques on the market cannot always claim to be the best. Sometimes tried-and-true veneers or crowns work best. Discuss your options with your dentist to select the best procedure for you.
Check with your dent. The situation is more important than the type of veneer. There are conditions were no preparation is required and there are other situations where preparation is not required. I would suggest seeing your dentist or a dentist with plenty of experience with different types of veneers evaluate your teeth and your esthetic goals then help you come up with a plan.