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methodcaller and attrgetter objects seem to be picklable, but in fact the pickling is erroneous:
>>> import operator, pickle
>>> pickle.loads(pickle.dumps(operator.methodcaller("foo")))
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
TypeError: methodcaller needs at least one argument, the method name
>>> pickle.loads(pickle.dumps(operator.attrgetter("foo")))
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
TypeError: attrgetter expected 1 arguments, got 0
When looking at the pickle disassembly, it seems that the argument to the constructor is indeed not pickled.
>>> import pickletools; pickletools.dis(pickle.dumps(operator.methodcaller("foo")))
 0: \x80 PROTO 3
 2: c GLOBAL 'operator methodcaller'
 25: q BINPUT 0
 27: ) EMPTY_TUPLE
 28: \x81 NEWOBJ
 29: q BINPUT 1
 31: . STOP
highest protocol among opcodes = 2