I was paid to build a software application as an independent contractor, but I did not sign a work for hire agreement. What type of lawsuit is the other party liable for if they use the software commercially without my consent?

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You would have to take them to court for copyright infringement. Unless you have a very good reason why your contract arrangement was not be a work-for-hire one and/or very good legal representation, expect the judge to quickly rule that it was work-for-hire and the copyright belongs to the company who paid you to make it.

Disclaimer: I am not a lawyer and this answer does not constitute legal advice.