We investigate measures of chaos and entanglement scrambling in rational conformal field theories in $1+1$ dimensions. First, we derive a formula for the late time value of the out-of-time-order correlators for these class of theories. Our universal result can be expressed as a particular combination of the modular $S$-matrix elements known as anyon monodromy scalar. Next, in the explicit setup of a $SU(N)_k$ WZW model, we compare the late time behaviour of the out-of-time correlators and the purity. Interestingly, in the large-$c$ limit, the purity grows logarithmically but the out-of-time-order correlators remain constant. Therefore, we find that some systems may display entanglement scrambling in the absence of chaos.