Dead head the dear things and keep digging up and digging up… So far as I know that is the only solution. Even we=hen you think you have totally cleared an area the seeds and bulbuls will still grow - just be totally ruthless!

Weedkiller works, but you have to add a little liquid soap to it to make it stick to the leaves long enough to be absorbed.
Been there, still doing it.

It's a tedious, back breaking business - you have to dig up a clump, put it on a tarpaulin or something, then break the clump apart and extract every single bulb and bulbil. Then repeat the process ad nauseum with every single spade or forkful... then dig over the ground that's left, and then tip back onto the area the soil you've got on the tarpaulin. And then do it all again the next year and the next, but if you've been rigorous enough the first year, you'll see a massive reduction in the second year and from then on, it's a case of never letting any stray ones escape each Spring. I've done this quite a few times with muscari and a white flowered allium, a bulb which is even more invasive than muscari.

As you have probably gathered its ongoing for many of us, good luck...

Gack! Sounds like what we have to do for Bermudagrass, here. As such, careful application of glyphosphate, sans diquat or suchlike additive, and plus a non-ionic surfactant, may be helpful in the larger, more open areas, but probably won't be a quick cure.

I have a small clump in my garden which I cherish every spring as they remind me of my childhood and the time my parents grew them. They are still a smallish clump after several years and show no sign of getting out of control. Perhaps I am doing something wrong?

Keep on doing it wrong, John ;-)).
Actually, we don't have a major problem with the muscari, it is the chionodoxa and ornithogalum that is more of a problem.

They're all a problem with us Bulba...

Bamboo - I managed to get rid of that white allium - just get the occasional one appear now

John - I think there's invasive muscari and non-invasive muscari

I think I must have the non invasive kind also( thank goodness). I did have a problem with Blue bells in one area, had to dig them up & carefully remove each bulb, some go very deep. Now it's just little seedlings that still come up. I snap them off. Some left in places where they're wanted, as do like them.

Well done Andrew on the allium - I still get the odd clump appearing, even after 8 years of rigorous attention - though they're usually beneath the foliage and amongst the roots of larger shrubs.