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If you land a spot in big law after graduating law school, then do you have to sign a contract to stay with them as an associate for 3 years, 5 years etc or hopefully it doesn’t work that way?

It does not work that way. You can work for them for however long you want. On the flip side, they can fire you whenever they feel like it for any reason except not an illegal reason. (It's at-will employment.)

Nope. Only thing is that they may try to claw back some of the bar/moving money if you leave before the year mark

No, its purely at-will employment. You can leave whenever you wish, and they can terminate you whenever they wish. You don't need to commit to a minimum amount of time there. And honestly, in many biglaw firms, they don't expect you to stay long anyway.

Of course, you may not be eligible for your bonus if you aren't employed on the date they are actually paid out (since most places make this a threshold eligibility criteria). Plus, as others mentioned, some firms will provide moving expenses or other incentives that they have the right to claw back if you leave too soon, though that depends entirely on the terms of the bonus agreement.

There's this weirdly pervasive idea among some associates that firms hate it when you leave after the 1 year mark. In my experience, basically no one cares if you do.