It's such a terrible topic but the whole death of a pet thing really does need to be discussed in advance so that, as others have said, plans can be made and requests noted well before the actual day it's needed.
On my pets notes at the vets, it is noted that I want a private cremation and ashes returned in a wooden casket. It's vital that my vets are aware of this in case ( a ) something happens when my pets are being looked after by someone else - if I am on holiday for example and ( b ) in case I am unable to voice my requirements at the time. I would hate for my pets to be....dealt with...in a different way but for it to be too late for my personal wishes to be met.
So although it's terrible to think about, it is well worth having your requirements noted by your vet, well in advance.
I have had the vet come out to our car for Stanley, Jessie and Sally. It was a single shot. Jessie had been given a sedative at home beforehand (planned a few weeks before and I had the medication stored for the day we needed to make That Call).
The others have been inside the vets and again, a single shot.
It's terrible. But its what we must endure.