Many of Copenhagen's sights are to be found on the waterfront, which is why tourists head not for the sightseeing buses but for the sightseeing boats. An hour's circuit round the harbour and the canals generally does it. [60 photos]
Boats are broad and flat to ensure they can pass beneath Copenhagen's low bridges, and generally open-topped to ensure everyone gets a decent view. Brand leader is Canal Tours, departing from the premier jetties, while Netto have less snazzy craft in not quite so obvious locations and will take you round for half the fare. Copenhagen's public transport offering also includes waterbuses, which although cheaper don't deliberately deviate past all the pretty bits and don't include commentary. On the positive side, you won't be urged to wave your arms in the air with a bit of a woo every time you go under a bridge. Disclaimer: we didn't ride on any of these, we walked round instead.
Most people start their tour in Nyhavn, the quarter-mile canal that's Copenhagen's picture-perfect tourist nexus. Both sides are lined by brightly colouredhistoric townhouses, repainted for maximum contrasting effect, while numerous sailing boats are moored up at strategic intervals. It's unbelievably attractive, or at least it is if you wander through during optimal illumination, but make sure you're standing on the waterfront not in the middle of the cycle path when you take your snap. Almost all of the buildings on the sun-facing side are cafes or restaurants at ground level, and three of them were once home to Hans Christian Andersen, who's very much the city's go-to celeb. [9 photos]
A separate canal loops round Slotsholmen, the island that doubles up as the seat of government, providing boatgoers with a properly sightseeingworthy circuit. The other loop threads through the heart of Christianshavn, a chain of artificial islands built within the harbour in the early 17th century, and that's more residentially attractive. But the chief water feature within København is the 'havn' itself, a saltwater channel separating the islands of Zealand and Amager which shrinks to 100m across at its narrowest point. Only three road bridges and four footbridges span the harbour, leaving plenty of room inbetween for messing around in boats.
The waterfront is also where Copenhagen concentrates its most modern buildings, safely separated from the more historic core. In 1999 the Royal Danish Library grew a boldcrystalline extension faced in dark glass and black granite, since nicknamed the Black Diamond, which extends almost to the water's edge. Pop inside to see the Museum of Photography, or lounge outside in a convenient deckchair. The neighbouring stack of glass boxes, Blox, isn't quite so visually-enticing but is somehow home to the Danish Architecture Centre. The other unmissable 21st century addition is Copenhagen Opera House, a vast auditorium with a knife-edge brim on the opposite bank, as yet surrounded by nothing much.
Walk far enough up the harbour and you reach the city's true global draw, the statue of the Little Mermaid. She was commissioned by one of the Carlsberg brewing dynasty, and can only be seen close up because she is truly little at just over four feet tall. The rock she perches on is only just offshore, so can be clambered onto by a mildly adventurous youth, which has led to a great deal of unfortunate vandalism over the years. Other tourists crowd on a drab stretch of waterfront to take their selfies, or are swung round in their sightseeing boats for a quick flash (for optimal accessibility, sit on the right). I'm glad I've seen her, but I hadn't been missing much.