We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus
E
S
ES
which parametrizes equisingular or topologically trivial deformations, the equigeneric locus
E
G
EG
which parametrizes deformations of constant geometric genus, and the equiclassical locus
E
C
EC
which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to
E
S
ES
corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to
E
G
EG
corresponds to the conductor ideal. We show that the support of the tangent cone to
E
C
EC
corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of
E
S
ES
,
E
C
EC
, and
E
G
EG
. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.