The history of taxes is a history of injustice. Originally taxes were imposed by the sovereign, chief, baron, war-lord or whatever you want to call the bully. Tax rate was whatever he could squeeze out of you. Lesser bullies in turn squeezed their subjects for whatever they could get and so on. Collecting taxes is like getting down from a goose, the object being to get as much as possible with as little fuss as possible.
Somewhere in history it occurred to the common people that taxes should be somehow proportional, and some of them fought for this small innovation. Theoretically the landless peasant or serf only paid what he could afford and the intermediate paid according to the same formula, but of course if you are destitute, giving up even 1% may mean not eating. The American Revolution was triggered by 1% tax! (Without representation.)
Proportional tax, has become a sort of standard, with the twentieth century innovation of progressive taxation. The idea being that for some 1% is oppressive and for the well off a much higher proportion is merely an inconvenience. The rub here is: What is a fair progression? As a result many tax systems have complicated algorithms that attempt fairness, but do not do it well. Even two individuals that appear to be equally taxable might have vastly different abilities to pay, one might have a sickly child, or land prone to flooding.
The first thing to consider is the tax base.
Equality: Everyone pay the same amount.
Disposable property: The tax collector comes to your place periodically and takes some of everything he can carry, or settles for what you donate.
Real property: The tax collector estimates what your property is worth, sends you a bill for a percentage.
Income: You and the tax collector figure out what your income was (will be) and then pay a proportion according to a complicated set of tables and calculations.
Flat tax: An income tax that is theoretically simpler and fairer because there is only one rate.
Consumption: Usually at point of final sale, Almost always a fixed rate in a given jurisdiction, but not applied to all goods and services.
VAT, GET and other hidden sales taxes collected at various points in the distribution chain.
Transaction: Just a sales tax on intangibles.
Excise: Relatively high tax applied to transactions the state wants to discourage, includes "sin"
Emissions: A tax on smoke!
Extraction: Based on what you take out of the commons.
Some taxes are imposed in a way that they are inherently regressive. The wealthy devote much of their income to things that are not subject to sales tax. Many transaction taxes are fixed per transaction so licensing a pound puppy costs as much as a $10,000 pure-bread lhasa apso. Social Security deduction tops put at $117,000 income.
The problem with many of these bases is evaluation. Two farms or fishing boats or barrels of whisky might look the same superficially, yet have vastly different values. Two identical boats might have different values depending on where they are. Two earners might have the exact same income, to the penny, but one lives in a city with high rents. Two coal fired smokestacks may appear to have the same emissions, but one is burning anthracite and the other lignite when no one is looking.
Some bases require businesses to collaborate with the tax collector, keeping detailed records of what was sold to whom at what location on which day in order to apply the correct tax rate to each and every transaction.
Flat tax sound good superficially, but as soon as one looks a little deeper the obvious limitation stands out: 15% of exactly what, gross income, take home, net, disposable? Cost of doing business has to be accounted for.
Many years ago I proposed a tax on energy, regardless of the source. My logic was and still is that energy consumption is proportional to wealth, and progressive also. Wealthier have more and bigger homes to heat and air condition. Bigger cars, yachts jets etc. They travel more, and more often in luxury accommodations. I learned that Al Gore proposed a carbon tax at about the same time. Narrowing it to carbon has some distinct advantages since carbon emissions are undesirable and most carbon comes from fossil sources that are finite. Energy measurement is inherently fair, A megawatt is megawatt whether it's from coal or oil, or gas, or nuclear. Now there might be a debate whether to include renewable or even nuclear, but that is a one time decision.
Taxing energy, or carbon at the source has distinct advantages in terms of collect-ability. Fluids or electricity can be metered continuously and counting railroad cars of coal is a lot less invasive than adding up cash resister receipts. Cheating an energy tax would be very difficult, you can't exactly hide a supertanker, a coal mine or a nuclear power plant.
Another advantage to extraction tax is that not only does it discourage the consumption of no- renewable resources and the inherent environmental damage is that it would encourage business to hire more employees where they can be utilized to reduce consumption.
I did the calculation a long time ago so I'm sure the numbers have changed a bit, but basically an energy tax equivalent 100% on a ton of carbon at the source could replace all other taxes, and all the record keeping and liability that goes with them..