Fingerprints

More than likely, there would be at least some partial fingerprints. He wouldn't be in any database though. They would also use luminol after they found the body and determined cause of death. They would find out where he was murdered.
Then if they got a search warrant, they could get his fingerprints, dna, etc. Both Connie and Edward would probably be suspects.

Yes! He left so many fingerprints and DNA everywhere..There is no perfect crime.

A bigger issue would be the rug fibers that would be in the trunk of Edward's car, if not actual some DNA from Paul as well. The movie was left with Connie and Edward's legal status unresolved, but if we consider that Paul was likely bedding many women, then Edward may not even have been enough of a suspect to get a warrant. Since most murders are committed by family members and there was no robbery, Paul's wife or her family would be bigger suspects.

I think the detectives were certain by Connie's odd reactions to their questions and to the news of Paul's death that she had an affair with Paul, but that isn't a crime (other than moral of course).

The body in the rub was buried in the dump for what was it like a week? That might have caused the prints to come off