The crazy cat lady is a prominent stereotype, but in the insane world of A Clockwork Orange, the catlady is one of the more sane characters. She runs some sort of health clinic and owns a ton of cats. She's the woman whom Alex murders.
The scene in which Alex kills her by crushing her face with a giant phallus is one of the more disturbing scenes in the film, which is saying a lot. Before Alex attacks her, he says to her,
ALEX: Naughty, naughty, naughty, you filthy old soomaka.
He's commenting on her sexually explicit artwork, even though he has similarly explicit art in his own bedroom, and his comment shows us that even though this world is sexually free, it's still male-dominated. The women are supposed to be the sex objects, not the ones in control. Alex beating her to death with a phallic statue drives this point home. The giant symbol of male sexuality literally kills a woman.