Zariski's moduli problem for plane branches and the classification of Legendre curve singularities

In this survey article we show several recent results on the local classification of irreducible plane curve singularities, plane branches, based on Oscar Zariski's lectures [17], and that of Legendre curve singularities. We compare exact classifications, the classification of plane branches and the classifications of Legendre curves, in the cases of simple and uni-modal singularities, and, as a consequence, we observe remarkable difference and mysterious similarity of these classification results. Also the comparison is achieved for a bi-modal case of characteristic (6, 7) which Zariski treated.