  An axial algebra is a commutative non-associative algebra generated by axes,
that is, primitive, semisimple idempotents whose eigenvectors multiply
according to a certain fusion law. The Griess algebra, whose automorphism group
is the Monster, is an example of an axial algebra. We say an axial algebra is
of Monster type if it has the same fusion law as the Griess algebra.
  The $2$-generated axial algebras of Monster type, called Norton-Sakuma
algebras, have been fully classified and are one of nine isomorphism types. In
this paper, we enumerate a subclass of $3$-generated axial algebras of Monster
type in terms of their groups and shapes. It turns out that the vast majority
of the possible shapes for such algebras collapse; that is they do not lead to
non-trivial examples. This is in sharp contrast to previous thinking.
Accordingly, we develop a method of minimal forbidden configurations, to allow
us to efficiently recognise and eliminate collapsing shapes.
