It depends on the specific sect and the specific situation. Natural methods like tracking the cycle and interruptus are pretty universally allowed. In general, Sunni are permitted to contracept on the condition that both spouses agree to it, it doesn’t result in permanent sterility, and it doesn’t result in harm to the body. So, basically no surgical sterilization and neither spouse is allowed to unilaterally decide to contracept. There is some debate about the last criteria and chemical birth control, so typically people will consult rulings of a mufti or advice by qualified Islamic medical professionals.
Beyond the three forbidden criteria, Muslims are instructed not to fear poverty and so many Muslims feel that there must be some sort of extenuating circumstance to warrant contraception - the risk of a dangerous pregnancy, for instance. Others feel that it’s okay to delay having children until a young couple is firmly established, out of school, etc, but in general Islam encourages people to wait until they’re financially solvent to marry anyway.
My subjective experience from listening to my aunties and cousins and talking to other people my age is that Western Muslims tend to prefer barrier methods and natural methods are more popular in the Middle East. One factor is that reliable barrier or chemical contraception can be quite difficult to get in many parts of the world, so that also affects what method people use. The more conservative a person is, too, the less likely they are to use an artificial form of contraception in my experience.