The UML 2.0 specification appears to be silent on the matter,
but why aren't stereotypes inherited along with attributes and
operations?

In fact, the default filter for stereotypes should be: show all
base class stereotypes
While we're at it, shouldn't stereotypes have visibility like
attributes? :-)
From the UML 2.0 Glossary:
[size=13]Stereotype:[/size]
A class that defines how an existing metaclass (or
stereotype) may be extended, and enables the use of platform or
domain specific terminology or notation in addition to the ones
used for the extended metaclass. Certain stereotypes are
predefined in the UML, others may be user defined. Stereotypes
are one of the extensibility mechanisms in UML.
While, the definition is correct, the implications of the
definition are not obvious, in my view... While the stereotype
is defining the extension of the metaclass (or stereotype), you
apply the denomination to an instance of the class... That is,
although the stereotype «enumeration» extends class, all the
definition says its that you may apply the stereotype
«enumeration» to a class. Not every class is an
«enumeration». Now supposing I make class X an
«enumeration», then it seems to me from first principles, that
all subclasses of X must also be «enumeration»s.
In addition, in my verbaliser technology, that I have
implemented in Rational Rose, I explicitly inherit stereotypes
and I have found it most useful in determining the validity of
the model. After you enforce stereotype inheritance, you may
find stereotypes appearing in what appear to be inappropriate
parts of the model. (Coupling this with stereotype adornments
that help them stand out)
When you investigate, you find that there is an ERROR in the
model! Fix the error, the inappropriate stereotype is removed
and all is right with the world!

Thoughts anyone?
Paolo