That isn't really the right question to ask. We never measure wave functions. We measure properties like position, momentum, energy of an electron. Whether the electron is spin up or spin down. The behavior of these properties doesn't match what you would expect from classical physics. Wave functions are a mathematical construct that help predict what measurements we can expect.
In classical physics, an electron is a small point-like particle. It follows a trajectory. A force acts smoothly to change the trajectory. You could measure position and momentum at any time you like to arbitrarily good precision without disturbing the trajectory.
By contrast, in quantum mechanics, the effect of the outside world on an electron is often better described by discrete interactions. We may know a measured value before hand. We can measure it again afterward. But we don't see what happens during an interaction. These kinds of interaction change the state of the electron, but they can tell us information about the the electron. We can use them to make measurements of the electron.
A typical such interaction would be to shine a light on an electron and learn things by how it is reflected. But light comes in lumps called photons. Bouncing a photon off an electron can tell you about the electron, but it also changes the electron's energy and momentum.
You can use a short wavelength photon, which can be well localized. This will tell you more precisely where the reflection occurred. But a short wavelength photon is high energy. It gives the electron a strong kick that can't be very precisely determined. So you don't know the electron's momentum or energy very well afterward.
You can use a long wavelength to make the kick as gentle as you like. But you don't know very accurately where this photon is. You can't learn as much about where the electron is.
This inverse relationship between accurately knowing position and accurately knowing momentum turns out to be fundamental, not just a limitation in measurement. It is one of the reasons why waves are used to describe reality. An electron does not have a precise position or momentum. It always has a range of possible positions and momenta.
These ranges are different from anything classical. A bag of gas doesn't have a definite size or shape, and is always spread out to some degree. Some parts of it can go fast and others slow. But everywhere inside the bag has some definite amount of gas and that part has some definite momentum. An electron is not like this.
You can shoot an electron through vacuum to a phosphor covered glass screen. If you prepare the electron in a spread out state, the electron has some presence everywhere in the vacuum chamber. You know this because it can hit anywhere on the screen with equal probability. But it is wrong to think that everyplace in the chamber has some piece of electron. When the electron hits the screen, it hits one phosphorus atom and make it give off light. The other atoms are not disturbed. If you repeat the experiment, you will find spots of light uniformly distributed over the screen.
It is also wrong to think that the electron just is a particle, and you learn where it is when it hit the screen. In the spread out state, it does not have a position. There is no way to predict which atom will be hit. If you put two slits in the way, the electron would go through both slits at once and would interfere with itself like a wave on the other side. You would still find one electron lights up one atom. But the distribution would would be concentrated where interference added and less where it cancelled.
An electron always has a range of possible positions. That range can be as big as a vacuum chamber or as small as an atom. It can be far smaller. No experiment has found a limit as to how small. But it cannot be $0$. If the position range is small, the momentum range is necessarily big. The electron in this state does not have a speed. There is no way to predict how long it will take to travel somewhere. It has a range of speeds and you can predict a range of times.
If you want to do physics with such an electron, you need to describe its properties and behavior with math.
Since the electron has some presence over an extended region of space, you describe it with a function over that region. The value of the function describes the "amount" of presence. The electron interferes with itself like a wave because it has a phase like a wave. So the value must have both a magnitude and a phase. Complex numbers fit. This function tells you all there is to know about the position of the election. The Fourier transform of it tells you all there is to know about the momentum. The function completely describes the state of the electron.
Further considerations about conservation of energy lead to the Schrödinger equation. This allows you to predict the form of the wave function in the presence of an electric field, or how it evolves in time. The Schrödinger equation is a wave function. The state evolves like a wave, and is called a wave function.
This only works between interactions. An interaction replaces the state with a new one. Once the interaction is done, the Schrödinger equation will tell you how it will evolve in time.
The Schrödinger equation predicts how the state of the electron will evolve as the electron crosses a vacuum chamber. It predicts that the electron will have a uniform presence across the screen. Since the electron only has a position spread out over the screen, neither the Schrödinger equation nor anything else can predict which atom will be hit. Once an atom has been hit and the electron has a new state, the Schrödinger equation predicts how the new state will evolve.
Quantum mechanics works very well, and has been experimentally verified many times many ways. But there are some obvious problems with it.
There is a perfectly good law that works as long as the electron doesn't interact with anything else. Something else must describe the interaction. And then back to the first law.
This just doesn't smell right. Quantum mechanics leaves room for "interpretations", which are mechanisms that explain the parts of the theory that can't be measured. The description so far has been the Copenhagen interpretation. Nobody has defined just exactly what an interaction is, or exactly what goes on when the unobservable wave function collapses.
People have thought about how to fix this. One way is the Many Worlds interpretation. In it, the wave function never collapses. It just continues to evolve.
When an electron hits a screen, the wave function encounters many atoms. Each atom has strong electric fields which affect the evolution of the part of the wave function near it. There are many states that the electron could enter, each describing the electron after encountering a different atom. The true electron state is a superposition of all of them.
The atoms all have wave functions. They have ground states and excited states. Each atom is strongly affected by the part of the electron wave function near it and weakly affected by the rest. Each atom enters a superposition of ground and excited states. There is a small amplitude that it is excited and a large amplitude that it is in the ground state.
The world splits into a superposition of many states. Each state evolves according to the Schrödinger equation and never interacts with the other states. In effect, the world has split into many worlds. Each world is complete in itself and is unaware of the others.
The Many Worlds interpretation has the advantage of being the most straightforward mathematical interpretation of the wave function. The drawback is having to accept that the world is continually splitting at an unimaginable rate, and we are just unaware of it.