  In the modular linear differential equation (MLDE) approach to classifying
rational conformal field theories (RCFTs) both the MLDE and the RCFT are
identified by a pair of non-negative integers $\textbf{[n,l]}$. $\mathbf{n}$ is
the number of characters of the RCFT as well as the order of the MLDE that the
characters solve and $\mathbf{l}$, the Wronskian index, is associated to the
structure of the zeroes of the Wronskian of the characters. In this paper, we
study $\textbf{[3,0]}$ and $\textbf{[3,2]}$ MLDEs in order to classify the
corresponding CFTs. We reduce the problem to a "finite" problem: to classify
CFTs with central charge $ 0 < c \leq 96$, we need to perform $6,720$
computations for the former and $20,160$ for the latter. Each computation
involves (i) first finding a simultaneous solution to a pair of Diophantine
equations and (ii) computing Fourier coefficients to a high order and checking
for positivity.
  In the $\textbf{[3,0]}$ case, for $ 0 < c \leq 96$, we obtain many
character-like solutions: two infinite classes and a discrete set of $303$.
After accounting for various categories of known solutions, including Virasoro
minimal models, WZW CFTs, Franc-Mason vertex operator algebras and
Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto
unknown character-like solutions which could potentially give new CFTs. We also
classify $\textbf{[3,2]}$ CFTs for $ 0 < c \leq 96$: each CFT in this case is
obtained by adjoining a constant character to a $\textbf{[2,0]}$ CFT, whose
classification was achieved by Mathur-Mukhi-Sen three decades ago.
