IMHO, bikes on multi-use trails need to be extra aware. The Fox River Trail (west of Chicago) can be very crowded in spots (StCharles-Geneva, etc) and requires a very slow casual pace on a bike. I've seen some real a-hole bikers screaming through crowded sections hollering obscenities at people because they're in the way. On quiet sections another danger is cruising along and not realizing you're about to cross a street and ending up in front of a car. Our annual WI trip (18th annual coming up) tends to avoid trails for these reasons (even though WI has some fantastic trails).
I prefer road drops because of the multiple positions (including the more aerodynamic "low" position for battling the dreaded wind). A good compromise is a cyclocross bike. They have mountain bike durability with road drop handlebars. And don't feel obligated to keep your drop bars super low.