Most ornamental forms behave very well. It is only the weedy, wild forms that can become rampant. The biggest offenders are wild allium (Allium ursinum), wild garlic (Allium vineale), and three-cornered leek (Allium triquetrum).

Chives (Allium schoenoprasum) can also reseed quite prolifically.

As per GardenGal: the species can be invasive, but many of the cultivars are certainly not.

Did have problems with Allium atroviolaceum planted in one garden. On the other hand, our Allium caeruleum just petered away. The A. thunbergii 'Ozawa' has behaved itself for years and the taller cultivar, Allium 'Purple Sensation', though very long-lived, has never presented any problem. Don't know know if the latter seeds, but it's large enough to conveniently deadhead.

Some of the bulb problems must also be to do with particular locations and growing conditions. For instance, have read complaints, on Houzz, about Muscari armeniacum (grape hyacinths). Personally, though cautious in using them, I've never found them to be runaways. In their case, I just have to be careful not to shovel them around.

Most Allium will spread around a bit from seed, A. aflatunense and karataviense are two that I encourage, the rhizomatous types (summer bloomers) can be a little too enthusiastic on sandy loam soil with many volunteer seedlings - I have grown A. thunbergii 'Ozawa' and 'alba' for the last 20 yrs and have never had a seedling from either, possibly because they flower in Sept - October here and don't have enough time to ripen seeds before the winter sets in.

If you don't want any seedlings from Allium simply cut off the flower heads when they are finished blooming.

I've never noticed the dwarf ones I have getting out of hand (they're either "Blue Eddy" or "Blue Twister" - can't remember which), but I haven't had them that long. I have had chives for as long as I can remember, and never noticed these reseeding much, either. Garlic chives is relatively new to me and still quite small, so not sure about that one.

Garlic Chives can reseed under the right conditions and come up so thickly it looks like lawn grass - so be aware!

I grew Allium nigrum (aka Allium multibulborum, old synonym)

wich gets leaves like a leek and really multiplies like mad.

there are still some plants around from bulblets that fell off when weeding.

personally I tend to loose my Allium aflatunense which just disappear, guess they needed more TLC/ fert or something

I've spent serious money on large flowered alliums that disappeared after a couple of seasons. I figured it was my fault so I bought more to try in different locations. No joy. Chives and garlic chives spread freely here. Purple Sensation still survives in a few very dry locations but doesn't spread.

I have a rather thick carpet of about fifty allium 'Millenium' seedlings, mabye will find a few variants among them. Honestly, I understand many are impressed with this plant, though I could take it or leave in favor of other alliums.

Well, I have to say Millenium is a trooper and it has offset nicely, but hasn't reseeded (not suppossed to anyway) in my yard over 4 years. I would say that I should divide my clump this year, it's gone from a strong gallon pot to about 4x as many bulbs.

I think it has been mentioned that 'Millenium' isn't sterile, though produces a lower quantity of seed and must love my cooler climate, I've got plenty of seedlings as well. 'Millenium' isn't something I'd run out and replace if I ever were to lose the thing, not that I would with all these seedlings going on.

Is it just me or has anyone else noticed a range of shades and even cultivars going under the name of 'Millenium'? The first time I saw it I swear the color was more saturated than what I saw for sale a few years later. Just recently I was at a friend's garden and it was obvious that half the 'Millenium' which was planted at a later date was a completely different plant.

Also the true Millenium is listed as sterile, but you never know if it's just waiting for the right pollen lol.

'Purple Sensation' is another one where you'll likely get a range of shades. I'm sure there are growers out there who deadhead relentlessly and rouge out seedlings or throwbacks, but I'm sure there are also ones who dig and sell anything with a nice bulb.