According to data, the verb mistrust is more frequently used than distrust. To mistrust is “to regard with mistrust, suspicion, or doubt.” The word suspicion can be key here.

Is distrust a verb or noun?

As a noun, distrust is the feeling of doubt.

Can distrust be used as a verb?

When do we use "distrust"? Both "distrust" and "mistrust" are most commonly used as nouns, rarely as verbs. So you will most probably find or use them as nouns, in contexts where somebody does not trust something or someone else.