There are exceptions, but not many.
There are several instances of unique d20 rolls, that are primarily related to spell effects or class abilities:
Wild Magic Sorcerer's Wild Magic Surge (PHB, p103) depends on a d20 roll for its trigger.
The Blink spell triggers its ethereal jaunt on the roll of a d20.
Duplicates from the Mirror Image spell are automatically targeted based on a d20 roll.
The Slow spell uses a d20 to see if a spell with a casting time of one action can be completed in the current turn or the next one.
In all cases I can find besides attack, saving throw, or ability check, the use of a d20 seems to be a randomized trigger, more than a check against or comparison to another value.