On conventional loans, Fannie Mae's homeowner resource page says conventional fixed-rate loans may only be assumed in the special situations it describes. In other words, a buyer should not expect to assume a conventional fixed-rate loan in an ordinary sale.
Whether any specific loan can be assumed depends on its loan documents and the servicer. The fastest way to get a reliable answer is to ask the servicer directly and get the answer in writing.
For FHA loans, HUD publishes the Single Family Housing Policy Handbook 4000.1, which includes sections on assumptions. An older HUD handbook chapter on assumptions states that FHA-insured mortgages are assumable and that, for mortgages closed after December 15, 1989, the lender evaluates the buyer's creditworthiness.
That chapter has a change date of March 24, 2011, so treat it as background and confirm the current requirements with the servicer. In practice, an FHA assumption is a qualification process, and the servicer can tell you what documents it will want.
For VA loans, the VA Home Loan Guaranty Buyer's Guide (April 2022 version) says anyone, even a non-veteran, can assume a VA loan, but that assumption requires servicer approval and in some instances VA approval. It also says servicers check the buyer's credit and income, and that the funding fee on an assumption, unless exempt, is 0.5 percent, a rate that VA's funding fee page also lists.
The guide explains that when a non-veteran assumes a VA loan, the seller's entitlement remains with the loan, while an eligible veteran can substitute their own entitlement, which allows VA to restore the seller's entitlement. For a veteran seller, that affects the ability to use a VA loan again, so it is worth asking about before agreeing to anything.
Other loan programs have their own rules, so ask the lender before assuming anything. If you are exploring VA financing more broadly, my guide to VA home loan buyers near Nellis AFB covers the VA process.