The highly esteemed and extremely proficient Thomas MaCurdy has a new piece in the JPE (jstor) on exactly that question. The news does not surprise me:

This study investigated the antipoverty efficacy of minimum wage policies. Proponents of these policies contend that employment impacts are negligible and suggest that consumers pay for higher labor costs through imperceptible increases in goods prices. Adopting this empirical scenario, the analysis demonstrates that an increase in the national minimum wage produces a value-added tax effect on consumer prices that is more regressive than a typical state sales tax and allocates benefits as higher earnings nearly evenly across the income distribution. These income-transfer outcomes sharply contradict portraying an increase in the minimum wage as an antipoverty initiative.

MaCurdy also writes:

About 35 percent of the total increase in after-tax benefits goes to families with income less than two times the poverty threshold, a common definition of the working poor or near-poor; nearly 13 percent goes to families principally supported by low-wage workers defined as earning wages at or below 117 percent…of the new 1996 minimum wage; and only about 14 percent goes to families with children on welfare. Unlike most public income support programs, increased earnings from the minimum wage are taxable. Over 25 percent of the increased earnings are collected back as income and payroll taxes…Even after taxes, 27.6 percent of increased earnings go to families in the top 40 percent of the income distribution.

File under “Scream it From the Rooftops!” I do not see an ungated copy, but here is an earlier 2000 paper (pdf) by MaCurdy, with O’Brien-Strain, with broadly similar conclusions.

From that same JPE issue, cream skimming effects seem to be pretty small when it comes to school choice.