Superpositions of quantum states exist in a multidimentional Hilbert space and they are not observable. Most every interpretation treats them differently - from ascribing a real world status(e.g. some forms of the MWI) to being nothing but our ignorance before measurement(CI).
The multitude of many wavefunctions form a Hilbert space and the Schroedinger equation determines how a particular wavefunction evolves over time. When you solve the equation and apply the Born rule, you always get probabilities for classical-like results('particles').