X-ray images are shadows of the tissue. Bones are really opaque so show up really clearly. Soft tissues, including cartilage, muscles, ligaments, fat and skin are almost totally transparent, so only show up super faintly.

The actual ligament itself only shows up the same as muscle - so when the two are next to each other, they just blur into each other because you can't see an edge where there is a difference.

However, you can sometimes see hints of a sprain. There may be swelling of the tissue around the ligament. While you might not be able to see any detail, you could see that things are swollen under the skin. Sometimes a sprain also damages the bone by pulling a small flake of bone off where the ligament attaches to the bone. The bone flake can be easily seen in many cases. The bones of the joint might have moved out of position of the ligaments holding them together have been damaged.