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I had to run an errand.
The Fool's Errand was created in 1987.
The Fool's Errand happened in 1987.
There is a Jazz lyric about an "errand girl for rythm." "Errand boy" would seem the parallel case. "Errand person," though, seems far too PC and stilted.
Yes, errand is singular; the plural is 'errands'.