If an IFR goes missed--as in not planned--I first issue our LOA missed approach so they're technically on that. THEN, if visual approach conditions (Ch. 7-4-2) are in effect I ask if they want vectors back for an approach or if they wanna stay with me in the pattern.
If they stay in the pattern and there's other IFR traffic I use tower and/or pilot applied visual. I then call approach and let them know they're staying with me and they're like, "yeah, nice". If there's too much traffic or a sequence inbound I don't do any of this but just send back to approach on the LOA missed to re-sequence. If they're GA I might ask if they want to cancel IFR to simplify things but often there isn't other traffic to worry about so remaining IFR is no big deal.
By and large air carriers aren't allowed to go VFR so I never ask; I keep them in the pattern if they want and it will work or I just ship them back to radar. Also, air carriers are often scared to just stay in the pattern so they get shipped back anyway.
Whoever said they become VFR after they miss is not correct. Like, if they go missed and it's IMC do they become VFR? Just because it's nice out and they go missed doesn't make them VFR.