To me, they look like "trace fossils". In other words, not a true fossilized plant or animal, but the path of a creature as it (slid, burrowed, walked) on (mud, sand, etc). Here is a google image search for trace fossils to give you an idea.
FC has as good an answer as I can think of. Other things like tree roots, insects and erosion or leaching of certain materials in the rock can form those types of lines as well.