The Scrappy: Polly. A lot of readers found her jealous behaviour (and the fact that Qwill forgave it every time) to be incredibly grating and unrealistic.
Seasonal Rot: The later books in the series were widely panned by both professional critics and consumer reviews, due to shoddy plotting and characterization. Many suggested that these installments were being written by a ghostwriter, though it was more likely due to author Lilian Jackson Braun's advanced age (she was born in the 1910s, but these books kept coming out until the mid-2000s!)
Values Dissonance: In The Cat Who Turned On and Off, Qwill finds an unpublished novel by one of the murder victims that brutally reveals the secret lives of his neighborhood's residents, which include a drug dealer, a bookie... and a "sodomite." Possibly slightly defensible as characterization, considering that Qwill, a middle-aged man in the 1960s with a somewhat working-class upbringing in the midwestern US, probably would not have especially sympathetic views toward homosexuality, but still jarring to a modern audience.