I, like the others in this thread, can only speak to Chicago laws. Here, a verbal lease is a perfectly valid lease, subject to the laws of the city and state (we usually call it "month-to-month"). I wouldn't say it's exactly common, but it's not exactly uncommon either. My last apartment was rented month-to-month for 4+ years (and was a nice apartment with a nice landlord).

In the cases I'm most familiar with, the apartment starts with a lease, and rather than renew the lease specifically, just shifts to a verbal lease, provided the landlord and tenant trust each other. (Were I renting an apartment to someone else, there isn't a chance in hell I would do it without a lease, though.)