This only issue I have with raising the minimum wage is that there are generally no provisions for raising wages of those who were already above minimum wage.
I saw this all the time when I worked retail. You start at minimum wage, work your ass off for two years to get a 50 cent raise, then the minimum wage goes up and new hires are making the same rate as you. It's demoralizing and makes you not want to out in any effort beyond the bare minimum in those jobs.
I'd much prefer if pay structures were defined as "minimum plus", where instead of being hired at $20/hour, you are hired at $5/hour over minimum wage (which is $15 in this example).
Of course this only actually matters to very low paid hourly workers in retail/food service/etc, and the corporations that control their lives will always find new and exciting ways to extract money from their employees