Elvis Costello, Royal Festival Hall

In a recent posthumously published article on the Guardian's website, the late rock scribe Steven Wells fulminated about a crop of musicians who, he believed, were way past their sell-by date. Wells was a terrific journalist, positively brimming over with sulphuric wit and he had a point with Bob Dylan. But he was well wide of the mark when he suggested that Elvis Costello should have been given his P45 in 1979.

In fairness a lot of the peaks were drawn from Costello's first decade, but onstage the artist formerly known as Napoleon Dynamite was an explosive force to be reckoned with, working through pretty much every musical style in his two-hour set, from Nashville to reggae via swing, blues, soul and a sultry Sinatra croon on "All or Nothing at All". The only omission was jazz. I guess he leaves that to his wife, Diana Krall.

Things did look a little iffy initially. He arrived with no introduction and ploughed straight into his set, as if he was about to make the notoriously grumpy Van Morrison seem like sweetness and light. On "Veronica", Costello’s 1989 collaboration with Paul McCartney, his vocals seemed to be dashing about all over the place in search of the right key. But then suddenly he hit his stride and proceeded to deliver the perfect balance of classics and new songs. He looked dapper in his tight dark suit and fedora and his singing was soon equally sharp.

Despite sounding cheerful, despondency seemed to be a motif among his latest compositions. "Down Among the Wines and Spirits" from his 2009 album Secret, Profane and Sugarcane, was about slipping so far down the bill your name is just above the liquor license. A bluesy lament from his just-recorded next album, "Jimmie Standing in the Rain", told the story of a cowboy music singer playing the Lancashire clubs just as cowboy music was going out of fashion. The philosophical 55-year-old might have been casting judgment on his own sales figures in these download days. As he ruefully observed after mentioning his new album, "I'm surprised you can still say 'record' without being taken away." But if he felt like a man out of time he was not going down without a fight. "Watching The Detectives" was breathtaking, as he contrived to play both the reggae rhythm and the film noir melody simultaneously thanks to a spot of nifty looping. "Radio Sweetheart" segued seamlessly into "Jackie Wilson Said", prompting a rapturous singalong. There was no sign of being on autopilot here.

It would have been nice to have seen Costello in full flow backed by The Attractions or his latest band The Sugarcanes, but it was an indication of the sheer muscularity of his back catalogue that stripped down to one man and his guitar they still retained their impact. “Good Year for the Roses”, from the 1981 album of country covers Almost Blue, was a fresh as a daisy and “Every Day I Write The Book” was Elvis at his literate pop best. It was hard to believe, as he claimed, that he had once fallen out of love with this sublime composition.

As the show reached its climax Costello was joined by Meltdown curator Richard Thompson for Thompson's bleak "End of the Rainbow" and "Shipbuilding", penned about the Falklands but still resonant today. The Costello of 2010 is a mix of raconteur, troubadour and political conscience – he recently cancelled gigs in Israel due to concerns about the Palestinian situation. But with his dad in the audience as a Father's Day treat this was a show that was never going to end on a downbeat note. The old Brinsley Schwarz feelgood stomper "(What's So Funny About) Peace Love and Understanding?" ensured that everyone left smiling. Maybe if the late Steven Wells was looking down on the RFH he might have grinned too.

• Meltdown festival runs until 21 June
• Buy tickets for Elvis Costello's UK tour
• Find Elvis Costello on Amazon
  
• See what's on at the Southbank Centre this season