1. Some of the workers accused Louis of stealing from the company, but they didn’t have a leg to stand on He had never stolen from the company and they had no proof that he had.
2. Stop trying to persuade me that you didn’t start the fight. You don’t have a leg to stand on All the children saw the whole thing, and they told me exactly how it happened.

Having a leg to stand on lends support to something, and not having a leg to stand on means support is lacking.