A few episodes have the murderer firing a silenced gun without any close by character hearing it. A silencer does not muffle the noise completely, as it can still be heard by someone nearby. (This is a common mistake in TV and films).
Columbo often undertakes actions which would result in evidence and investigations being compromised, these actions could result in cases being dropped or overturned upon appeal. There is no way that the District Attorney would allow Columbo to perform the actions that he does in the show.
In multiple episodes, Columbo manages to get a "confession" out of his murder suspect(s) in their own home and proceeds to tell them how he found out the truth. It would be a gross violation of police procedure to reveal the evidence against the suspect outside official police interrogation and possible plea deal.
The description of the 'Mrs. Columbo' character is not fully consistent with details from the mentions of her in many episodes.
There can be a few plot holes realized in certain episodes. With the way the murderer is covering up the murder and how Columbo figures it out.