One of my favorite types of cop characters is what I call the "Columbo Ex Machina" character. This is a character who lurks deceptively, often dismissed by the protagonist as useless, and then pops out to save the day at the end. You most often see this kind of character in romantic suspense. He may even be the hero, working secretly in the background and arrives in time for the rescue, despite the heroine's best (and misguided) efforts to thwart him.
This character can be even more fun, though, if it isn't a romantic lead. I think the perfect example of this is Lt. Nathan Shapiro, the sad-sack NYPD detective created by Frances and Richard Lockridge. While the later Shapiro stories took on a more police procedural style, the first few were suspense stories in which Shapiro seemed to play a minor role. (Two stories of this kind are "The Drill is Death" and "Murder and Blueberry Pie.")
The protagonist of the story would be locked in a life or death struggle, perhaps even accused of a crime he didn't commit. In the background, Shapiro would schlep a long, in his own depressed way, one step behind the bad guys. He was so mild and easy to dismiss. He even dismissed himself. The other cops would be fooled by the villain's plot, and Shapiro, when he found himself dissatisfied with the case, assumed that he must be wrong. Those other cops were so much sharper than he was.
But little details would bug him, so he would slowly and sadly follow it up. He was a great device for building suspense because the audience knows he's right. They want him to just keep poking, even though all the forces in the universe are pushing him to drop it. And meanwhile the hero or heroine would have no reason at all to believe this sad, discouraged little man would be their savior. But then, at the end, he would show up, gun in hand, sadly apologizing for taking so long about it. It was like being rescued by Droopy, and it was very satisfying.
Of course, this kind of character doesn't have to be a policeman. Agatha Christie used this model sometimes with Miss Marple - just letting her sit in the background, knitting, while the story followed other characters who were desperate and mystified. I think this is actually a good model for refreshing a familiar character. The audience knows exactly what the character is capable of, so a little goes a long way, and the fact that the protagonist may not realize just how valuable this person is, well, that can build a lot of anticipation.
And speaking of Miss Marple - tomorrow I'll talk about that bastion of British policework, Inspector Slack.
(The other entries in the series are The Dismissive Policeman, Policeman As Nemesis, and Policeman as Community Member. )