Their uses align with the meanings of the individual kanji, basically.
現れる means that something that was hidden shows itself. Imagine the sun appearing from behind clouds.
表れる means that something internal comes to the surface. Imagine an expression on someone's face.
顕れる means that something (usually something bad) becomes clear, or is discovered. For instance, someone's wrongdoing.
現れる is the most common by far. You are exceedingly unlikely to run into 顕れる.
As with many words with nuances that can be expressed by a variety of kanji, sometimes people just opt for the most common one, even if it's not technically the correct nuance. They really shouldn't be interchangeable. It should always be fairly clear which one is most appropriate.
I took these definitions and examples from this monolingual dictionary entry.
https://www.weblio.jp/content/%E3%81%82%E3%82%89%E3%82%8F%E3%82%8C%E3%82%8B
現れる、表れる、and 顕れる
Is there any semantic difference between 現れる、表れる、and 顕れる、or are they completely interchangeable?