Riding a bike (or even a motorcycle or moped) is always dangerous in a scenic touristed area - everyone, including you, is looking at the scenery! I once gave a lift to a young Canadian tourist who crashed his motorcycle along Diamond Head Road because he was looking at the ocean on the right as the road deviated left. He and his feet were badly scraped up and his planned trip on to New Zealand was probably not much fun. And add to that the fact that the traffic in Honolulu is the worst in the nation - including Los Angeles - and you'll understand why it would be best to stay on 4 wheels.

As for mountain bike trails, I know of none, and if there are any, I would stay off them. The mountains on all the islands get very steep, and hikers frequently have to be rescued, sometimes just "retrieved". On most ridges and scenic rises, the terrain doesn't have a granite or dense lava base, but rather, a gravelly/cindery base that crumbles and slides, and without safety lines, even hiking is dangerous. My advice is to stay on foot or 4 wheels and to stop before ogling intently.

~Reggie~