Ben Casselman has a good piece on this question, here is one excerpt:
During the strong labor market of the mid-1990s, only 1 in 5 minimum-wage workers was still earning minimum wage a year later. Today, that number is nearly 1 in 3, according to my analysis of government survey data. There has been a similar rise in the number of people staying in minimum-wage jobs for three years or longer…
Even those who do get a raise often don’t get much of one: Two-thirds of minimum-wage workers in 2013 were still earning within 10 percent of the minimum wage a year later, up from about half in the 1990s. And two-fifths of Americans earning the minimum wage in 2008 were still in near-minimum-wage jobs five years later, despite the economy steadily improving during much of that time.
The piece is of interest throughout.