Tearjerker: Trout really goes through a number of hardships in the narrative. Finally recognized, he's mugged in New York City, has his feet hardened in plastic after wading through the polluted river, unwittingly inspires a demented man to attack other people, and loses a piece of his finger when he tries to intervene. To say nothing of the conclusion, where he meets Vonnegut and begs him to make him young again. It's further cemented by the final illustration of the novel: Trout, with a single tear rolling down his face.
While most of the novel's many, many tangents are played for laughs, Vonnegut going off on how his mother killed herself with sleeping pills, and how her suicide affected his fight with depression, can really hit the reader in the gut.
The Woobie: Trout, for the reasons listed above. Creeps into Jerkass Woobie territory, when he acts in a hostile manner, to illustrate his lack of success and recognition in the arts.
Patty Keene also counts. She was raped before the events of the novel, and her attempt to turn her life around is foiled by Dwayne's mental state keeping him from recognizing her plea for employment.
Jerkass Woobie: Dwayne Hoover. The novel spoils that he'll turn into a psychopath and hurt several people, but he's still a poor man damaged by his wife's suicide. The late-stage syphillis doesn't help, either.