  We extend the noncommutative residue of M. Wodzicki on compactly supported
classical pseudo-differential operators of order $-d$ and generalise A. Connes'
trace theorem, which states that the residue can be calculated using a singular
trace on compact operators. Contrary to the role of the noncommutative residue
for the classical pseudo-differential operators, a corollary is that the
pseudo-differential operators of order $-d$ do not have a `unique' trace;
pseudo-differential operators can be non-measurable in Connes' sense. Other
corollaries are given clarifying the role of Dixmier traces in noncommutative
geometry \`{a} la Connes, including the definitive statement of Connes'
original theorem.
