To find someone's weakness, fault, or evidence against them; to catch someone out or get a lead on them (idiomatic).
刑事はついに犯人の尻尾を掴んだ。
The detective finally got a lead on the culprit.
彼の嘘の尻尾を掴んだので、上司に報告した。
I caught him in a lie, so I reported it to the boss.
証拠が見つかれば、会社の不正の尻尾を掴めるはずだ。
If we find evidence, we should be able to get a lead on the company's wrongdoing.