What do I do with grape hyacinths when they have flowered?

I think the answer is ....nothing. Let them all die down naturally, tidy up when they have done that and they will come again year after year.

Nothing. I agree. Unless they are in the wrong place for you. In that case let them die right back, dig them up and replant them elsewhere. You will also have plenty to share with neighbours.

We do dead head them or they seed absolutely everywhere.

Hmmm, I'm afraid I know what I'd do with them, lol.
Darned things get everywhere.

Thankyou. I will give them a bit of a tidy up cos they look a bit of a mess. I'm sure I'll be back soon. I've got a lot to learn.