These short stories have serial killers, either as one of the main characters or as someone being pursued by the authorities. A relevant anthology you might like is Psychos: Serial Killers, Depraved Madmen and the Criminally Insane. See also:
“12:01 AM” (A Karen Vail Story) by Alan Jacobson
Vaughn is on death row and due to be executed in 2 hours. He thinks back on his “work” and accepts his fate. He takes some solace in the fact that his legacy will live on. Three hours earlier, Debra Mead was coming out of a grocery story when she was pulled into a van. A rookie detective noticed a similarity to an old case.
This story can be read in the preview of Nothing Good Happens After Midnight: A Suspense Magazine Anthology. (28% in)
“The Bully” by Jeffrey Deaver
Henry is in the Eagle tavern when he spots his bully, Stan Whitcomb, a very big man. He wants to get out of there, and hopes his friend Larry won’t notice any problem. Henry has complained to the Sheriff’s Department about Stan and now he has to again after Stan confronts him.
This story can be read in the preview of Exit Wounds. (22% in)
“The Commuter” by Jeffrey Deaver
Charles is getting the train to work when he gets a call from his wife. She’s worried about him because another body was found—a victim of The South Shore Killer. He’s not concerned and is annoyed by the call. His loud talking disturbs the other passengers. He also calls Carmen, his mistress and a competing company about selling insider information.
This story can be read in the preview of More Twisted: Collected Stories Vol II. (54% in)
“Chip Assassin” by Mark Gardner
Josephine and Doug investigate the scene of a murder. The potato chip killer has struck again, leaving a sour cream and onion this time.
This story can be read in the preview of Stories on the Go: 101 Very Short Stories by 101 Authors. (87% in)
“#8” by Jack Ritchie
A man and his young redheaded passenger listen to a news report about someone who’s killed seven people. The police are searching the area and have roadblocks set up. The young passenger is impressed with the killer, and talks about his motivation and how smart he is.
This story can be read in the preview of Alfred Hitchcock’s Mystery Magazine Presents Fifty Years of Crime and Suspense. (65% in)
“Collecting Ackermans” by Lawrence Block
Some buzzes Florence Ackerman’s intercom. A man says he has a telegram from Western Union. Being suspicious of criminals, Miss Ackerman challenges his identity, turns him away and threatens to call the police. The next day, as she walks home from some grocery shopping, she’s grabbed and pulled into an alley. Soon after, Freitag and Poolings, police officers, talk about the case. It reminds Freitag of something.
Some of this story can be read in the preview of Enough Rope. (89% in)
“A Good Man is Hard to Find” by Flannery O’Connor
An extended family is headed to Florida for a vacation. The grandmother wants to go to Tennessee instead, so she talks about an escaped murderer—The Misfit—who is suspected to be on his way to Florida. Despite her efforts, her son Bailey is set on going to Florida.