A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$ with a smooth $1$-dimensional critical set $\Sigma=\{(x,y)\in \CC\times \CC^n \mid y=0\}$. An isolated line singularity is defined by the condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line singularities were classified by Dirk Siersma and are analogous of the famous $A-D-E$ singularities. We give two new characterizations of simple isolated line singularities.