The most common crimes committed in the great outdoors are vandalism and littering. Nature is the number one victim, but when it's your backcountry it tends to feel like a personal attack. There are some crimes committed specifically against people. Petty theft does happen, but you're just as likely to have your stuff trashed as you are to lose it by theft.
For the most part people get along in the backcountry, when you hear about confrontations it's usually between hunters; either disputes about who shot an animal, who has the 'rights' to shoot a particular animal several people have been tracking, or people getting upset because someone scared of their prey or they're in someone else's hunting grounds. They don't typically escalate too far, mostly because everyone is carrying a gun, but fist fights aren't unheard of.
There are other disputes as well, in my home town, in the winter, you've got two groups of people: the skiers, and the sledders (snowmobilers). They have a mutual agreement over who gets which valleys to play in, really the skiers only have one valley that they restrict to skiing only, they groom and maintain a double track nordic ski trail that goes back to a couple cabins deep in the woods. The sledders have their onlyown valley that they play in, and groom for snowmobiles, and the two clubs respect each other's terrain and stick to their own trails.
However, a number of years back there was a new guy in town who went exploring and tore up the ski trail on his sled. He go caught by one of the prominent ski club members, who happens to be closer to 7ft tall than he is to 6ft. There are a couple versions of the story, but my favourite is the one where the large viking of a man clothes-lines the sledder right of his sled, words are exchanged, and the sledder ends of getting decked in the face and laid out in the snow. Shortly after that, the ski cabin got torched.
This is the kind of vandalism people are most worried about, people breaking into other peoples trapping, sledding or ski cabins deep in the backcountry, vandalized them, or making a mess and leaving garbage all over the place.
I will offer a word of caution to tourists who venture into the backcountry outside of parks or publicly maintained trails: The locals take great offence to outsiders helping themselves to their trails and 'facilities'. If you aren't in a park, then all the trail improvements, maintenance, and structures built in the back country are done by the locals, which takes a lot of time, effort, and money. They aren't intended for public use, or abuse, and you risk a good thrashing for 'trespassing' (even though legally it isn't, but there isn't any law enforcement out there, so...). Show respect, and you will receive respect.