  Repeated local measurements of quantum many body systems can induce a phase
transition in their entanglement structure. These measurement-induced phase
transitions (MIPTs) have been studied for various types of dynamics, yet most
cases yield quantitatively similar values of the critical exponents, making it
unclear if there is only one underlying universality class. Here, we directly
probe the properties of the conformal field theories governing these MIPTs
using a numerical transfer-matrix method, which allows us to extract the
effective central charge, as well as the first few low-lying scaling dimensions
of operators at these critical points. Our results provide convincing evidence
that the generic and Clifford MIPTs for qubits lie in different universality
classes and that both are distinct from the percolation transition for qudits
in the limit of large onsite Hilbert space dimension. For the generic case, we
find strong evidence of multifractal scaling of correlation functions at the
critical point, reflected in a continuous spectrum of scaling dimensions.
