  We study the structure of singularities in the discrete Korteweg-deVries
(d-KdV) equation. Four different types of singularities are identified. The
first type corresponds to localised, `confined', singularities, the confinement
constraints for which provide the integrability conditions for generalisations
of d-KdV. Two other types of singularities are of infinite extent and consist
of oblique lines of infinities, possibly alternating with lines of zeros. The
fourth type of singularity corresponds to horizontal strips where the product
of the values on vertically adjacent points is equal to 1. (A vertical version
of this singularity with product equal to $-1$ on horizontally adjacent sites
also exists). Due to its orientation this singularity can, in fact, interact
with the other types. This leads to an extremely rich structure for the
singularities of d-KdV, which is studied in detail in this paper. Given the
important role played by the fourth type of singularity we decided to give it a
special name: taishi (the origin of which is explained in the text). The taishi
do not exist for nonintegrable extensions of d-KdV, which explains the relative
paucity of singularity structures in the nonintegrable case: the second and
third type of singularities that correspond to oblique lines still exist and
the localised singularities of the integrable case now become unconfined,
leading to semi-infinite lines of infinities alternating with zeros.
