This is probably too enormous a question to answer properly - the only real answer is "police work."
Ted Bundy, for example, was arrested twice (he escaped after his first arrest) in traffic stops. The first time, the arresting officer noticed some weird equipment in his car - a ski mask, handcuffs, pantyhose. The second time, his plates turned up as belonging to a stolen car. He was also a chronic shoplifter.
Other killers developed patterns of antisocial behavior, getting into the same kinds of altercations with neighbors, kidnapping victims in the same kinds of places and then moving them to another location to which previous victims had been moved, lighting fires....
Many times - really dismayingly many times - the police had testimony from victims who'd escaped, but without enough evidence to arrest the killer. The most horrifying of these, in my opinion, took place after your 1986 limit, when a teenaged Laotian immigrant escaped Jeffrey Dahmer's apartment in Wisconsin after being drugged and lobotomized, was found wandering on the streets and was returned to Dahmer's custody by police who a. didn't know about Dahmer's prior conviction for sexual assault, b. had trouble understanding what the kid was saying, and, in all likelihood, c. didn't want to get involved in what seemed to be a homosexual domestic dispute.
I don't believe DNA evidence was that pivotal in Dahmer's eventual conviction.
Dahmer, like other serial murderers, tended to keep trophies - in his case, body parts soaked in acetone or kerosene or just stuck in a freezer, the kinds of things one wouldn't need DNA to identify as human. (With Dahmer, neighbors or relatives smelling foul odors was a recurrent theme - he'd keep bodies around for days and use chemicals to reduce them to disposable form.)
John Wayne Gacy buried victims under his second home. Ed Gein made things out of his victims' bodies (including a pair of female private-parts shorts he'd wear). Charles Ng and Leonard Lake made videotapes of their murders. That seems to be part of the psychology of the serial killer.
So once warrants were issued and a search could be performed, there were plenty of "smoking guns" to be found... in many cases. Not all.
Often, killers under arrest will lead police to more bodies than were being investigated.
Often, too, I think it's fair to say that DNA has been more useful in identifying victims after the killer has been caught. Maybe someone with more criminal justice experience can contradict that, but it's my impression.
If you want to read more, the FBI hosts "vaults" of info on people like Dahmer and Gacy.
Most of what I've read, though, comes from various news stories published at the time of arrest and trial - I used to have access to some pretty hefty clip files, not all the components of which ever made it into LEXIS or the web. Check the archives of a local paper during the sentencing phase of a trial if you want details on any given case.
One of the books offering a more unusual insight into Gacy and the serial mentality was Jason Moss' The Last Victim - he was a teenager who'd befriended Gacy, Dahmer and a few other killers behind bars and didn't quite realize what he was letting himself in for.
It's sensationalized - Moss was really obviously seeking attention, but also had a unique level of access to Gacy especially. (Unfortunately, that meant Gacy had a unique level of access to him, too.)