I would be very hesitant to provide it.
Social security numbers were invented for the social security system, to track employee contributions to the fund during their working years, and manage disbursement of benefits once they retired. The only other legitimate uses are in other matters related to taxation and earnings: e.g. filling out your income tax return, arranging for income tax withholding when you hire on at a job, etc. No one else really needs it. Other entities use it simply as a convenience, because it is a unique identifier for the majority of folks who have one.
The only possible use a university would have is to cross-check your income tax records as part of a financial aid application. Unless financial aid impacts actual acceptance to the university, though, you should not need to provide it on the application. You probably will need to provide it if you are applying for financial aid, as someone else mentioned. I suppose, then, you could wait until you receive notification of which uni accepts you, and provide your SSN only to them, for processing the aid application. If universities do this in parallel, though (consider aid applications before notifying applicants of acceptance) you will need to provide it now.
Bottom line: rather than guess, simply call the admissions office at the unis you are applying to and ask them if you really need to provide it just to apply for admission. Explain that you are concerned about giving it out especially with the growing problem of identity theft, and want to understand better what they do with the information. This is a legitimate question/concern and I doubt you would be the first applicant raising it. See what they say, and then decide.
As for medical offices, none of them need your SSN, unless perhaps you are at a VA clinic, or using another federal benefit (Medicare/Medicaid might need it). I NEVER give my SSN to such places. Some of them try to tell me that they can't process my insurance claim without it. I tell them that none of my medical providers have ever had that trouble, but there is always a first time, so if they do have trouble, let me know right away. None has followed up with me on that.
I've never filled out an application form that didn't ask for it
