My wife and I were once attending a play with friends. When the play was over, we were walking out. They have a son who is friends with my oldest son. We happened to be walking by Dr. Gary Smith, MD (a national specialist in children's safety) of Nationwide Children's Hospital. When they asked if we thought trampolines are safe, I introduced them to Dr. Smith and repeated the question. Dr. Smith said "No, do not buy one or let your child use one. They are dangerous". I said thanks and kept walking. That in a nutshell is how pediatricians feel about trampolines.
There is a good review of this here at Shots the NPR Medical Blog. The American Academy of Pediatrics has again released an updated statement about trampolines and is available here. We know there is a "thrill" factor that attracts everybody to trampolines. And we have kids that jumping on a trampoline keeps them active when they may struggle to be physically active in other ways. But we urge you to have your children avoid trampolines, even those with safety features and when limiting the trampoline to one person at a time, to avoid the serious and not-too-serious injuries that can occur with their use.