  We show, in this first part, that the maximal number of singular points of a
normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically
closed field $K$ of characteristic $2$ is at most $16$. We produce examples
with $14$, respectively $12$, singular points and show that, under several
geometric assumptions ($\mathfrak S_4$-symmetry, or behaviour of the Gauss map,
or structure of tangent cone at one of the singular points $P$,
separability/inseparability of the projection with centre $P$), we can obtain
smaller upper bounds for the number of singular points of $X$.
