The US qualifications may vary slightly form state to state but in California is is based on minimum income on a quarter basis over the past year. This threshold is pretty low but the benefits are minimal also. You can continue to work part time but the earnings are subtracted from the unemployment benefits so rarely a net gain.
The real problem with part time jobs is that they usually don't include health insurance benefits. It is cheaper for an employer to give one full time worker health insurance benefits instead of giving two part time workers health insurance benefits.
I think if the US had free national health insurance it would really change the landscape of part time jobs as more people would be open to it.
