We had a short stocky dragonfly today with a pale blue tail. I've just
looked it up and it's Libellula depressa, which is a darter. I then saw
what I think must have been the case it emerged from half way up the
leaf of a water iris. I should have been more observant and spotted the
case before, and then I could have been aware of it hatching!

I thought that they spend their nymph stage in the water and then crawl
out when it's time for the adult to emerge, so I'm not sure where the
egg-laying fits in.

They do spend their nymph stage in the water, the libellula you mentioned
(broad-bodied chaser) lives in the mud at the bottom of the pond for a
couple of years. they then emerge, spend a couple of weeks away from the
pond maturing. Then they come back to mate, lay their eggs, and the whole
cycle begins again.

The big ones I have in my pond (33 emerged now by the way) are Anax
Imperator (Emperor dragonfly) - the nymphs usually take 2 years to mature
and are the size of a newt - pretty fearsome - with voracious appetites,
they will attack anything in the pond that moves!