From Bloomberg:
There were 10,080 households with adjusted gross income of at least $200,000 annually, or 0.26 percent of high earners, who paid no income taxes in the U.S. or to other countries. Using an expanded concept of income that includes some nontaxable items and looking only at U.S. income taxes, there were 35,061 households, or 0.88 percent, that paid nothing.
Methods of avoidance include tax-exempt interest and deductible charitable contributions.