There are different ways that people try to gain knowledge about what happens after death. Usually, Muslims look to scripture (Quran and hadith) for information about life after death.
The majority of Muslims interpret the Qur'an and hadith to say that people are only born once and die once (apart from the possibility of the raj'ah, as held by Shi'is, and which is not reincarnation). The standard view among Sunnis and Twelver Shi'is is that people do not reincarnate or transmigrate, and every being has its own soul.
Beyond that, it is hard to say with full certainty what happens after death, since we cannot learn about life after death through conventional means.
If it were possible to prove what happens after death with full certainty in a way that all people would agree and accept it, these things would not be questions. No one disagrees about what happens immediately when you arrive at Los Angeles airport, because it is known and provable. Life after death is a different sort of subject.
While some people argue for reincarnation on the grounds that some young children seem to remember things about previous lives, there are also other non-physical explanations for how they could have this sort of knowledge, so it isn't necessarily proof for reincarnation.
What one can say that, if there were a possibility that people reincarnate, the Qur'an did not consider it important to tell us about it. Instead, it tells us to focus on this life and how we are living right now, and not to guess whether we might be suffering due to wrongdoing in a previous life.
It also tells us to interact with others equally and fairly, based on the idea that they have come to this life newly, rather than what one might guess about their previous life.
Sometimes, belief in reincarnation can lead people to limit themselves or treat others wrongly, due to the belief that they are bearing a karmic burden or something equivalent, and this is against the Qur'anic ethos of justice.
As with all things, God knows best.
