I'm pretty sure mine has been stolen a dozen times at this point. You should never assume your SSN is private information, but you should treat it as such to limit how many people have it.
The main issues here are:
- applications for credit - freeze your credit at the major credit bureaus - Experian, Equifax, Transunion (bonus points for ARS and SageStream); make sure to unfreeze if you apply for a credit card or bank account though
- impersonating - like applying for jobs and whatnot; this shouldn't directly impact you, and it's on the employer to make sure they know who they're employing
- password resets - the best you can do is use MFA, though many services will allow resets with just personal information; I hope this changes, and some orgs are doing things to prevent abuse (e.g. Fidelity has voice recognition to cut down on support scams)
Honestly, we really need to stop using the SSN as identification.