  The hypotenuses of all right triangles inscribed into a fixed conic $C$ with
fixed right-angle vertex $p$ are incident with the Fr\'egier point $f$ to $p$
and $C$. As $p$ varies on the conic, the locus of the Fr\'egier point is, in
general, a conic as well. We study conics $C$ whose Fr\'egier locus is singular
in Euclidean, elliptic and hyperbolic geometry. The richest variety of conics
with this property is obtained in hyperbolic plane while in elliptic geometry
only three families of conics have a singular Fr\'egier locus.
