It's probably not so much the existence of a contract, but the contents and specificity of any contract.
In the US, most employees below VP or director, will have very generic employment contracts. I've been at my employer for 20+ years and haven't signed a contract since I was hired, even through one merger and several PE acquisitions. I haven't signed anything about my normal salary either - they send me a notice every year when raises are granted and my reading it is treated as acceptance.
My bonus is loosely based on company financials, and subject to board and executive approval. As far as I know, the pay-out schedule isn't set in stone either. I have no doubt that if the company failed, I wouldn't see a penny of any unpaid bonus.
I'm not sure that's true for executives, who often have individual contracts with more specifics about bonuses, whether they be cash, stock, or something else.
Such is life in an at-will state.