  This paper presents a general definition of pseudo-differential operators of
type $1,1$; the definition is shown to be the largest one that is both
compatible with negligible operators and stable under vanishing frequency
modulation. Elaborating counter-examples of Ching, H\"ormander and
Parenti--Rodino, type $1,1$-operators with unclosable graphs are proved to
exist; others are shown to lack the microlocal property as they flip the
wavefront set of an almost nowhere differentiable function. In contrast the
definition is shown to imply the pseudo-local property, so type 1,1-operators
cannot create singularities but only change their nature. The familiar rule
that the support of the argument is transported by the support of the
distribution kernel is generalised to arbitrary type $1,1$-operators. A similar
spectral support rule is also proved. As no restrictions appear for classical
type $1,0$-operators, this is a new result which in many cases makes it
unnecessary to reduce to elementary symbols. As an important tool, a convergent
sequence of distributions is said to converge regularly if it moreover
converges as smooth functions outside the singular support of the limit. This
notion is shown to allow limit processes in extended versions of the formula
relating operators and kernels.
