Should Dogs be Allowed on Public Trails?

This afternoon I headed out to a local preserve where dogs are allowed off leash in one flat mowed area of the park and on some of the trails. In the rest of the park, dogs are required to be leashed but some people don't follow those rules and consequently many dogs run free on all of the trails. This doesn't appear to be a problem because most everyone who visits this park comes there to run their dog off leash.
But what happens when a non-dog lover visits this park only to be jumped on by an overly friendly dog or worse to be met by an aggressive dog? What if said person has a young child in tow that gets knocked down or steps in a "present" left by a dog. On the other hand, what about the creek beds that suffer from erosion or habitats that are destroyed by loose dogs?

As a dog owner, I want to let my dog be free to run but also want to respect the natural lands we visit. Should dogs be allowed on public trails? For me, the answer would be that we need to set aside a few dog friendly trails in every community and sign them accordingly. This would give dogs a place to visit with other dogs in a free environment while restricting the negative impact dogs could possibly make to just a few of the trails within a natural preserve. It's OK to have rules regarding usage of parks and dogs, just don't make them too restrictive or unfair.