Q: Someone accidentally took my umbrella from the coat rack and left his. May I take his umbrella?

A: Although he mistakenly took your umbrella, you may not take his. Instead, you should post a sign there that an umbrella was mistakenly switched. This applies even to mitzvah items, such as a tallis, siddur, etc. (C.M. 136:2; Kessef Kodashim ad loc.).

If the owner comes and gives simanim but claims that he did not take your umbrella, you must still return his (Pischei Choshen, Aveidah 4:19).

If no one comes to claim the umbrella, after a reasonable time you may use it, as explained in previous questions regarding items whose rule is yehei munach. Additionally, if the owner is unaware that he took the wrong umbrella and continues to use yours, you are entitled to use his umbrella in lieu of rental for yours. However, this would not apply to using a switched tallis, since we require full ownership to be able to make a bracha (Hashavas Aveidah K’halachah 7:10).

In a public place, where this frequently happens, the administration can institute and publicize a practice that people not be particular (see Aruch Hashulchan 136:2).