Nobody warned me about taxes in my first year. April did. And April is the worst possible teacher, because by then the lesson is already expensive.
Here is the basic picture. Stipends used for living expenses are generally taxable income in the US. The tuition waiver part usually is not, but the part that pays your rent is. I know. It feels wrong. The tax code does not care about your feelings.
Withholding is the gotcha that gets people. Some universities withhold taxes from stipends like a normal paycheck. Others do not withhold at all, which means you owe quarterly estimated payments, which means you are your own payroll department. If nobody told you this, you are not alone. Almost nobody gets told this. Departments treat it as obvious. It is not obvious.
Once you know your real monthly take-home, a grad school budget planner app becomes the difference between guessing and knowing. Map the after-tax number against rent, groceries, and everything else. Then use a grad student expense tracker app to check reality against the plan each month. The plan is the easy part. The tracking is where the truth lives, and the truth is usually that you spend more on food than you think.
Fellowship stipends and assistantship wages can be taxed differently in the details, and international students get another layer of treaty rules on top. I am not a tax professional and this is not tax advice. The universal takeaway is simpler: ask the department how stipends are paid and whether taxes are withheld before you sign anything. Get it in writing if you can.
Do your monthly math on the after-tax figure, not the headline. That is the number you will actually live on. Then compare that take-home against local costs, because a stipend is only as good as what it buys. The per-program figures at phdstipend.fyi give you the gross numbers to start from. Run your own tax estimate on top, and you will know what each offer is really worth. Future-April you says thanks.