Qualifications
For a fee, you can get certified by the Mystery Shopping Provider's Association. If you join MSPA, you can register to take the silver or gold certification test.
You don't have to be certified to get mystery shopping jobs and yes, certification does cost money. You're paying for training on how to be a mystery shopper and since you're an independent contractor, you can deduct the cost off of your taxes like your other expenses.
However, if you don't have the money to join, you can still work for an MSPA company. It just might take you longer to get the higher-paying assignments. All things being equal, there's still no guarantee you'll get any more work if you join or get certified.
Some people can figure out the job on their own, just by reading the documentation provided from the mystery shopping companies that they register with (and you'll need to register with many of them to cobble together a "career").
You can also find companies to work for by going to the MSPA website, where they list over 400 legit shopping companies. Or you could look up companies individually, free of charge, and register with those companies directly.
Additionally, SASSIE is a mystery shopping search engine that will match you with companies such as Intellishop, Beyond Hello, and others.
Once you register with a mystery shop service, you usually have to fill out an application that shows off your writing style, emphasizing how detailed you can be when reporting on a typical shopping trip.
Then you'll have to learn about the different clients that each mystery shop service handles. You'll need to spend time reading documentation about the companies and what they're looking for with the actual shops. Finally, you'll have to take online tests so that the client is certain you know what to look for in a shop situation before they'll take the chance of giving you the account.