Alan Cumming has an abject fear of singing in public – which is understandable given that, in truth, his singing voice isn’t terribly strong. But in his new one-man cabaret show, I Bought a Blue Car Today, you get so much more than songs. In the extended links between the 17 numbers, arranged by musical director Lance Horne, Cumming proves himself to be a raconteur of the highest quality – engaging, amusing, insightful and self-deprecating – who grows in confidence as the evening builds.

There’s a good smattering of showbiz anecdotes about an eclectic mix of personalities, of whom Cumming is by turns either surprisingly starstruck or tragically unaware - from Graham Norton to Cyndi Lauper, Mika, Walter Cronkite, Whoopi Goldberg, Mikhail Baryshnikov and, most hilariously, MGM musicals legend Ann Miller – and amusing experiences with paparazzi chases, the Hollywood merchandise machine (did you know that Cumming is a Flintstones’ chewable vitamin?) and gossip websites.

And while he doesn’t attempt to trace a complete autobiographical timeline, Cumming mines enough gold nuggets from his career to keep fans happy and to prompt others less well informed to marvel at just how much he’s accomplished. Theatregoers in particular will be thrilled by his behind-the-scenes view of performing at the Tony Awards, instruction on the RSC’s “surprised and delighted” curtain calls and his dusting off of “Mein Herr” from Cabaret, which he dedicates, with touching understatement, to the late Natasha Richardson who played opposite him as Sally Bowles on Broadway.

It was Sam Mendes’ Donmar production of the Kander and Ebb musical that first took Cumming to the US, where he’s been largely living for the past decade and has now become a citizen (the show’s title refers to a sentence he had to write for the English-language portion of his ‘naturalisation’ test). His observations about his adopted country are wry and refreshingly honest – and, in what feels like a joyous homecoming, the Scot makes clear his enduring allegiance to the UK, where amongst other things, “gays are treated with respect”.

Of the songs, Cumming succeeds in bringing the most panache to comic numbers like Victoria Wood’s “Thinking of You” (in which he also demonstrates his skills as a pianist), an expletive-ridden encore number called “Beautiful” and “Next to Me”, a joyfully smitten tribute to his lover, which he co-wrote with Horne. Elsewhere, there are interesting choices – but the delivery is often under-powered and feels forced and too “actorly”, particularly compared with the easy and informal nature of his conversation.

In some ways, though, it doesn’t matter one jot. While I Bought a Blue Car Today may not be very exciting musically, as an intimate snapshot of a complex and charming man, it’s fascinating – and absolutely unmissable.