What studies have found works well are 2 things:
1) Sales taxes. But you have to have a system in place that refunds estimated sales taxes to people below a certain income threshold (the closer you get to the threshold the less you get back), and you can't tax food. Similar to the GST system Canada has. If you're poor you pay nothing (or even negative) federal sales tax. If you're rich, well, you just paid 20% on that 5th house you bought, and 20% on that 19th car, and 20% on the incredibly expensive high end faux fur coat. If you import something you also have to pay sales tax on it, or this system doesn't work.
2) Capital gains taxes. They shouldn't be too high (don't want to discourage investment), but they should be progressive. The more of your income your derive from capital gains, the more you pay. This should work like income tax: everything up to X amount: no tax. Everything between X and Y 20% tax. Everything between Y and Z 40% tax (etc). (So if you made 1 dollar initial tax bracket you'd only pay tax on that one dollar.)
And you also have to make it so that tax paid decreases the longer you hold an investment, down to a minimum amount. (Hold for 1 year: Goes from 40% to 35. 5 years: 30.) This is to discourage rapid buying and selling, and encourage holding long term stocks.
You also have to tax people on a percentage of their gains even if they're planning to reinvest their money. Once they sell a stock, you tax them on (some small percentage though) of their gains even if they reinvest within a short timeframe, or 100% if they pull the money out. This is to make sure you aren't giving rich people a completely free ride on most of their returns. "Oh, their broker reinvested that last billion they made minutes after selling? Well, looks like we'll only tax them on the 2 million they actually took out of the market this year!" When you're a billionaire you can let almost all your money ride indefinitely, so having full tax exemptions (as some places do, to encourage investment) for letting your money ride is stupid.)
When you have two systems like that in place it somewhat helps stabilize lower income people. Tax reform has to work in conjunction with other programs, of course.