Hard to get rid of. Same with me when we moved recently. As the weather has been so mild I have been digging up all the plants with their bulblets attached, seems to be working but it's a long slog.

Oh dear, there isn't an easy answer - Cammomile's method is the only one that works, and even then, you need to repeat the exercise each spring, because inevitably, small bulbils get missed and remain in the soil and grow the following year. Best to dig out anything that looks like wild garlic as soon as it appears each year.

Muscari are the same. I never did manage to stop them taking over in my last garden. I refuse to have them now pretty though they are.
Chives ditto. Best kept in a pot and not allowed to flower as they will make little bulbs ANYWHERE.

When the ground is soft, I find a fork is better than a spade as you are less likely to cut through a bulb.

Make lots of spaghetti! ;D
Seriously, I don't know of anything better than digging, for weedy plants of that type. Glyphosphate, mixed with a surfactant, may work, but it is easy to apply too much, and then have to deal with residual toxins. Also, that shouldn't be applied too close to anything that you want to keep!

Andrewr - just out of interest, what do you normally dig with, spade or fork? Unless I'm dealing with a completely empty area, I always dig with a fork, always have done, employing the spade at certain times, but mainly with a fork, because the soil's heavy here. Doesn't work where the soil's light and sandy, but on heavy soils, its a fork for me every time, just wondered what you do....

Bamboo - as my garden is so heavily planted, there's not much opportunity for digging. However I do prefer a spade when space allows. The fork is useful when soil needs 'sifting' (as in removing small bulbs), also for lightly adding garden compost or feed.

Ah, well that tells me your soil there isn't as heavy as what we have here - there's no question of 'sifting' the soil with a fork, it comes up in great clods... Half the time, when I need to dig a planting hole, I don't even need the spade, the soil comes up in such large lumps. I did once live in a different part of this area of London and there the soil was more or less alluvial silt, sand and grit, could have dug it with a tablespoon...

Well I have been here for thirty years and the soil has been improved during that time.