Using the classification of transitive groups of degree $n$, for $2 \leqslant n \leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 \leqslant n \leqslant 48$. The results of this classification are made available as a supplementary database.