As a purchaser, you would often want fixed-rate. A variable rate is subject to the ups and the downs, but a fixed rate you can always refinance (at a cost, but one that's beaten out in the long-term by better average interest on a typical loan) to a lower fixed rate when available and ignore when rates climb. Since 30-yr loans are common, plans that play out on 5-10yr timescales are relevant. Given that banks are willing to offer them, you'd need an out-of-the-ordinary situation (selling soon, believing rates will continually decrease slowly, not being well educated, unique credit situation making rates favorable for one loan type and not another, ...) to choose a different option.
This is definitely country-dependant. In Sweden you can pay extra with no penalty if you're on a variable rate, so if you have the means, the best option financially speaking is to go with variable and pay it off as quickly as possible (or at least enough to make the raise in interest to be negligible).
Most mortgages in the US can also be paid off with no penalty - fixed or not. A purchaser should prefer fixed if they are available, and only use a variable rate loan for specific circumstances. The low rates of the last few years compressed the rate difference between fixed and variable, that it made almost zero sense to get a variable rate loan.
In the US some can have prepayment penalties, but it's not common. All the loans I've ever been presented (car, house, etc...) were simple interest loans. I would never sign one that was anything other than simple interest.
