Depends on the animal. Some make more useful trails than others.
Caribou, for instance, seem to favor the approach of making a million minor trails on every hillside. Easy to find, useless to follow.
Much better if bears are involved. Bears are both large and smart, so they make pretty good trails. I guess I usually find them where it makes sense - along rivers and lakes, at constriction points in the landscape, etc... They usually serve to make it easier to get through the brush, but occasionally have led me on a useful shortcut or around an obstacle. They can sometimes also be good for linking up meadows, in a place with patchy meadows in brush.
Then there are the "only way" trails. These are most often mountain goat trails, in really steep areas. The mountain goats have often found the only way through that doesn't require climbing gear. Bear trails can do this too, occasionally. I once followed a bear trail that actually required a couple foot jump across a gap in the trail - and that trail was the only way across a really steep slope.
And I haven't even mentioned all the vole trails, ground squirrel trails, etc...
My animal trail thoughts mostly apply to Alaska, though. Down here in Washington even many off-trail places have relatively fewer animals and relatively more people. So a lot of the game trails follow more human rules.