Zero. I'm not sure how you really get stuck. Only thing I can think to maybe use it for is maybe if I have to work myself into a corner decorating to get it to look right.
Once. My net broke while I was high on a mountain cliff and I was too lazy to walk to the shop to buy a new one, so I let rescue services carry me there.
I used it once to see if using it once would give you Nook Miles. I thought it was worth checking because there ARE a lot of features in-game that do this. It did not so have not used it since.
I almost use it but only for the nook miles + thing where you need to redeem 500 nook miles. Getting your NM deducted from the call but then cancelled counts as redeeming nook miles so ¯\_(ツ)_/¯