Galanthus
Here in central Indiana, we have lots of clay soil. I have tried to naturalize Galanthus in the lawn. Mowing is what eventually eliminated them, but I think not the clay soil. When planted in beds -- i.e., anywhere without grass -- they do well. I grow GG. nivalis, elwesii, and a few woronowii. They do just fine, away from grass and lawnmowers.
Note however that many species of Galanthus are simply not hardy in our climate, regardless of the soil. I've stopped experimenting with Galanthus and Cyclamen, since they are difficult to come by, and I got tired of killing them. Still, GG. elwesii and nivalis do really great here.
Lycoris, Cochicum, Sternbergia
So do Lycoris in addition to squamigera: chinensis, caldwellii, longituba, and sprengeri.
Hardy Cochicum like byzantinum, cilicicum, speciosum, and a few others are terrific here, even in the grass (if you don't mow till their leaves yellow off). My neighbors love the pink flowers in our lawn in later summer! They never complain about the "hay field" effect the unmowed patches of lawn give into early July.
Others that may seem unlikely but that do well include Galtonia, Sternbergia, most Eurasian Gladiolus, and a few hardier Crinum in protected spots (but full sun). You can't have too many Sternbergia! They bloom after the Colchicum have finished flowering.
The Lasagna Method
That said, it is not a good idea to just stick everything into plain clay soil. Add gypsum. Add sand. Add lots of composted leaves, or just pile your leaves on top of the beds when you rake them in autumn.
I make beds on top of the clay, starting with a very thick layer of rotted leaves; then put on a 2 to 4 inch layer of some decent topsoil; and finally cover with 2 inches of plain brown sand. Plant in the sand/topsoil layer (they get mixed when you try to plant anything). The roots will go down to the rotted leaves layer all on their own. I call it the "Lasagna Method."
No spading, but a bit of hauling and spreading. Outstanding drainage! But the clay is close enough that the roots can go down into it, if they need to, to find water in dry weather.
I didn't invent the lasagna method; I got the idea from one of these on-line plant groups. Try it; you'll love it.
Can You Grow Bulbs in Clay?
So, can you grow bulbs in clay? In many case the answer is yes, but you really don't have to. Use the "Lasagna Method" instead.
Good gardening, from here in central Indiana
Jim
Look up technical terms in the Glossary of Plant Biology