I'd really like to see an experiment conducted that can put the whole debate to rest: raise the minimum wage in half of the states, lower it in the other half. Run this experiment for three years and compare the results, and apply the strategy with the best outcome to the rest of the country.
It looked at contiguous US states, where one raised the minimum wage and one did not. It measured the gain or loss in jobs with all these "natural experiments".
This study found minimal effects on employment when comparing the states that raised minimum wages against those that did not.