Can a music school sue an instructor for "breach of non-compete obligation" if the instructor never signed a contract?

There is no guarantee that your son's former employer will not sue, even if it doesn't have a very strong case.  Many employers aggressively litigate against former employees who compete against them, knowing that they are more able to afford the costs of litigation than their former employees. While the fact that your son never signed a non-compete agreement is certainly in his favor, it is not dispositive.  Non-compete or not, people cannot use confidential information they acquired during their prior employment to compete against their former employer.  In some cases, such protected information could include a customer list or other customer information.  Also, the former employer may claim that your son orally agreed to a non-compete.  While such an agreement is required to be in writing if it can't be fully performed within one year, and non-competes are in general carefully scrutinized by Courts and often held to be either fully or wholly unenforceable, an oral non-compete agreement for less than one year may theoretically be enforceable. In sum, from the little information I have, I think your son has a pretty good position, but it is not a "slam dunk".  However, even if the former employer was to successfuly sue him, they could potentially have him enjoined from using their trade secrets and/or violating an oral non-compete agreement until it expires, and collect damages, but I don't see how they could take his right to teach away, compel him to pay the school's legal fees, etc.