Several MSPs appear to be undergoing current IRS audit to verify that they are using folks with correct SSNs on file. There are very few legitimate companies that do not ask for a SSN as part of the registration.
So here is how many of us handle it. IF it is a company we know to be legitimate from other shoppers' experience (as mentioned in forums such as this, or companies with enough shopper experience that they are on the lists of legitimate companies here, at Volition.com or MSFreedom.org) I am willing to give them my social security number. If it is a company I have heard nothing about, I am more cautious about giving out the information. Some companies will approve you without a social security number, some won't. If you aren't approved you won't have an opportunity to work with them.
When you register, look at the little address bar at the top of your browser. This should give you an indication that you are or are not signing up on a secure website. If there is the little padlock, I will go on and enter my number. If there is no padlock I will attempt to register using 000-00-0000 or 999-99-9999 in hopes that after I am approved I can go to "my profile" on their website and enter a true and correct SSN in a safer environment.
Your SSN is already in the hands of way too many people--your school, your bank, your credit card company, your cell phone provider, your insurance company, etc. etc. There is no way of absolutely preventing identity theft from someone determined to commit it. Giving your number to a legitimate mystery shopping company probably does little to increase your risk. If it is a major concern, you may want to get identity theft insurance. It is available through some banks, through AAA and other organizations. While probably unnecessary, if it makes you sleep better at night it may be a worthwhile investment.