I have both. I have my tenors fastened at the heel and my concerts fastened at the headstock. I own a couple of sopranos, without straps. I play them only sitting down around the house, so I don't need a strap. I can strum simple first position chords without a strap while standing, but anything very complicated requires a strap and I stand when performing. My straps stay on my ukes.
I have bought web belts at flea markets to use as straps. For the concerts, I braid leather boot laces and tie them at the headstock.
I don't put my head through the strap, but hang it on my right shoulder. I play mandolin this way, so when I took up the ukulele, I did it this way too.
Kala with braided leather boot lace and reso with web belt
Mandolin and tipple with South American woven belts
Strap over right shoulder. I must have been doing
Ukulele Lady or
Let's Talk Dirty In Hawaiian, since I'm wearing a lei.
How do you wear your strap? I put my head through for heavier instruments like guitars, but not for ukes and mandolins.
(Sorry 'bout the "attached Thumbnail". I posted it by accident and can't seem to get rid of it)