Vexed, BBC Two

Lucy Punch – what a great name for a comedian (or a female boxer). Unfortunately that is the only thing that’s great about Vexed, a new comedy drama written by Howard Overman, creator of Channel 4’s perky ASBO (RIP) superpower fantasy Misfits. His new show is that relative rarity, a comedy cop show, a genre of which Punch has some experience, having had a supporting role in Hot Fuzz, although it’s not in Vexed’s interests to start making such comparisons.

I mean, Vexed isn't vexatious; it trots along perfectly agreeably – there were no noticeable longueurs in last night’s opener – it’s just that I didn’t once come near to breaking into a laugh, or even to raising a smile. “Bright, sexy and funny” is the BBC’s description of the show. Bright? Passably. Sexy? Did I miss something? Funny? Like I said...

So what did we expect in August? Well, I’m afraid that Sherlock and Grandma’s House have raised the bar this summer (I have to disagree here with my theartsdesk.com colleague’s review of Simon Amstell’s sitcom, which I thoroughly enjoyed) – and the BBC appear to have a policy now of chucking in one or two golden nuggets amongst the seasonal landfill. We’re not all off on holiday, goes the new enlightened thinking. And, like I said, Vexed isn’t exactly bad.

The opening scene held promise, as a couple were shown round a flat by an estate agent. Only this flat was a murder scene, the couple were a pair of detectives, and the corpse was still lying on the living-room carpet. Nasty stain, says the female detective, Kate, who thinks she might pick up a bit of a bargain. You can always hide it under the sofa suggests her male colleague, Jack.

Darkish, deadpan humour - a little bit Green Wing, a little bit Naked Gun – is the predominant key then, but deadpan requires the straightest of straight faces, and neither Lucy Punch (Kate) nor Toby Stephens (Jack) – a down-at-heel version of Hugh Grant – is anywhere near the Leslie Nielsen class. After all, deadpan requires its protagonists to carry on as if they weren’t in the middle of a farce, while Punch, in particular, is given too many reaction shots. And if anyone displays signs of sanity it rather dilutes the lunacy.

Anyway, last night’s story involved Kate and Jack attempting to catch a murderer who is using a supermarket loyalty card scheme and customers' purchase histories to pick out victims. However, while Kate used the case to find out whether her husband is having an affair, Jack decided to use the database in order to pick up a woman he had bumped into in a supermarket. Predictably, they both get it wrong – Kate knee-capping her husband over the mistaken belief that he has been buying roses and saucy underwear for a lover. Like many of the gags, you didn’t need binoculars to see this one coming.

This is a three-parter (as was Sherlock), and whether it can improve over the run will depend on the chemistry between Punch’s Kate and Stephens’s Jack. Kate/ Punch doesn’t seem to have much to work with, unfortunately – Jack being a thoroughly one-dimensional character. And I can’t see there being any will they/ won’t they tension between the couple because, on any rational analysis, Kate wouldn’t touch Jack with a barge pole. Okay, so in the dog days of August, Vexed might just about pass muster. But unlike the case of Benedict Cumberbatch’s Sherlock Holmes and Martin Freeman’s Dr Watson, I can’t envisage the internet forums being full of clamouring pleas for the return of this particular pair of detectives.

• Watch Vexed on BBC iPlayer