  We study the singularities of a modified lattice Korteweg-deVries (KdV)
equation and show that it admits three families of singularities, with
analogous properties to those found in the lattice KdV equation. The first
family consists of localised singularities which can occupy an arbitrarily
large domain but which are, nevertheless, always confined. The second family
consists of one or more lines extending all the way from the south-west to the
north-east on the plane, involving a single finite value that depends on the
parameter that appears in the equation. We argue that the infinite extent of
this singularity is not incompatible with the confinement property or with the
integrability of the equation. The third family consists of horizontal strips
in which the product of values on vertically adjacent lattice sites is equal to
1. In the case of the lattice KdV equation this type of singularity was dubbed
`taishi'. The taishi for the modified lattice KdV equation can interact with
singularities of the other two families, giving rise to very rich {and quite
intricate} singularity structures. Nonetheless, these interactions can be
described in a compact way through the formulation of a symbolic representation
of the dynamics that is similar to, but in a sense, simpler than that for the
KdV case. We give an interpretation of this symbolic representation in terms of
a Box&Ball system related to the ultradiscrete mKdV equation. These results
show that taishi-type singularities are not limited to the lattice KdV
equation, but might very well be a general feature of integrable lattice
equations with deep connections to other facets of their integrability.
