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