Strictly speaking, a flat tax is the border between progressive and regressive taxation and is neither progressive nor regressive.
Ideally you would want to exclude the expenses incurred to sustain a reasonable minimum quality of life and only consider taxation on income in excess of that. A truly flat tax rate is still regressive in the sense that it is a significantly higher burden on that expendable income at the lower end of the scale.
(A lot of analysis gets based on the misuse of $ as a flat measure of utility, which is deeply flawed.)
EDIT: more accurately, this subsumes and generalizes the disposable income issue, rather than ignoring it, since the concept of "necessities" that come out before income is "disposable" is really a simplification of the "low hanging fruit" of high-utility spending to a binary category, rather than a continuum.
I don't agree that is ideal. Ideally, everyone makes an equal sacrifice. What is an equal sacrifice?
* Everyone could pay the exact same amount of tax, let's say $15K. But that is not equal sacrifice; some people can afford $15K much more easily than others (especially those who don't have $15K).
* Everyone could pay the same percentage of their income, let's say 17%. That also is not equal sacrifice: If you earn $1 million, you can afford to give up 17% of it much more easily than if you earn $10,000.
* Tiered progressive tax rates: These can be equal, if calibrated effectively. It will never be perfect and always up for debate, but it can be done pretty well.
1) Figure how many times poverty rate you make in revenue (2x, 3x, whatever)
2) Take the log base 10
3) Multiply by some flat constant that is the same for everyone, so that the total national amount taxed is how much the US needs. Last I checked, that flat constant would be around 9.
4) The result is what percentage of your revenue you should pay in taxes.
Currently, we are far from logarithmic. Under this scheme, pretty much everyone reading this board would pay less taxes. Bill Gates pays about the right amount. Exxon would pay more.
Useful as a thought experiment to explore how far away we are from a fair level of progressive taxation.
$5000 means more to a working class person than a billionaire. If taxation should take an equal amount of value from each, then a flat percentage is comparatively regressive.