1 Ny Carlsberg Glyptotek
As the name might suggest, this wonderful art gallery was named after the famous Danish brewery. Founded by the brewer Carl Jacobsen (1842-1914), it includes some of the most important examples of art from ancient and modern times, including 35 works by Gauguin and several extraordinary Van Goghs. The building's focal point is the winter garden, where Danish sculpture and Roman sarcophagi stand amid palm trees under a great glass dome. A curvaceous sculpted marble figure lounges in a pool while 20-odd cherubic children clamber around her. Overlooking this idyllic scene is an award-winning restaurant (Dantes Plads 7, tel:  3341 8141; www.glyptoteket.dk).
2 Bohemian Quarter
It's hard to get lost while shopping in Copenhagen: the centre is small, and the department stores (Illum and Magasin) and great fashion outlets (particularly Norgaard pa Stroget) line the pedestrian street known as Stroget. But it is when you venture into the smaller side streets that things become really interesting. Pisserenden, or the bohemian district, is situated northwest of Gammel Torv, and many of the basements have been turned into studios by young fashion designers. Tone Barker offers flowing outfits in devore silk, jersey and satin (Vestergade 11); Uno specialises in comfortable clothing in a kaleidoscope of colours (Vestergade 13); and Laedersmeden (Vestergade 15) sells quality leather footwear. Vintage-clothing enthusiasts will also have a field day in this area. One of the best second-hand shops is Kobenhavn K (Studiestraede 32B), where everything is washed and ironed before being put on display. If you get peckish, Cafe Istanbul (Larsbjornstaede 25) sells pizza slices from as little as 12 kroner (HK$15.70), and the shawarma kebabs filled with slithers of marinated beef are a delight. Then, at the end of the day, have a bottle of cool ale in the low-key Elefanten og Musen (Vestergade 21), where an open fire, candles and a couple of wooden tables create hygge (comfort and cosiness) in the true Danish sense of the word.
The Danes love cycling as much as the Chinese do. And, fortunately, Copenhagen is made for it. All the main streets have cycle paths, and there are special racks where you can pick up a free City Bike, as you would a shopping trolley. Your 20-kroner coin is refunded when you return the bike to any of the 120 stands around the city centre (www.bycyklen.dk). Copenhagen is also a great place to buy a bicycle, and there are many unusual cycling shops in the bohemian quarter and Christiania. Sogreni has designs reminiscent of the Pashley and Dutch classics, although a special zinc coating and subtle paint finishes give them a modern edge. Irresistible details include flat mudguards in wood or various metals, handlebars with grips made of leather and rattan, and dinky bells made from a simple disc embossed with the company's logo. The company is working on the prototype for a wooden bicycle, and other experiments include a fur-clad behorned two-wheeler named the Mountain Goat (Sankt Peder Straede 30A; www.sogreni.com). The Pedersen - the Bentley of the classic cycle world - can be found at Christiania Bikes in Christiania. The frame of this extraordinary bicycle is made from 14 metal tubes welded together to form 21 triangles, creating a kind of Eiffel Tower in motion. Meanwhile, peddling parents will love the bikes designed with a trailer in front for transporting children, or anything else (Refshalevej 2; www.christianiabikes.com).
Besides the bike shop, Christiania is a fascinating glimpse into an alternative world. The streets are ramshackle, and the ageing army barrack buildings are well past their prime, but if you can see beyond that, there is plenty to discover. Established as a kind of social experiment in the early 70s, the free town has been tolerated by an otherwise conformist society ever since, despite sporadic government efforts to move on the residents. The people of Christiania pay no rent for their accommodation, only water and electricity, and all along Pusher Street hash dealers sell their wares, quite openly and without fear of arrest (Prinsessegade, Christianshavn. For guided tours, tel:  3295 6507). Less controversial are the great places to eat. Spiseloppen gets rave reviews (Badsmansstaede 42, tel:  3257 9558), and vegetarians and vegans will love Morgenstedet, originally set up in the 70s for musicians to provide them with their early morning post-gig replenishments (Badsmansstraede 43, www.morgenstedet.dk).
5 Louisiana Museum of Modern Art
It may not be Copenhagen central, but this unique museum is a must. Situated on the waters of Oresund (the Sound), 30km north of the city, Louisiana takes about half an hour to reach from Hovedbanegarden, the main station. Here, an exceptional collection of 20th-century art is displayed both indoors and out, but it is the building - all wood, windows and light - as well as the beauty of the setting, that make the experience special. Founded in 1958, Louisiana is set amid the ancient trees of a great park. Wooded slopes, a picturesque lake and the sea beyond offer glorious views from inside the gallery. The parkland is dotted with sculptures by the likes of Henry Moore and Max Ernst, while indoors, delights include a collection of Giacometti's twiglet figures, fanciful scenes by Chagall, Van Gogh's luscious oils and the pop art of Andy Warhol. Louisiana also offers peaceful walks, concerts, films, plays, seminars and workshops for children, so there's definitely no risk of museum fatigue here (13 Gammel Strandvej, Humlebaek, tel:  4919 0719; www.louisiana.dk).
6 Rosenborg Castle
This former royal residence, a beautiful Dutch Renaissance style castle, offers the ultimate treasure hunt, with the crown jewels housed in the underground vaults. Built as a summer retreat in 1603-34 for Christian IV, and used by successive monarchs up to Frederick VII (who died in 1863), over the centuries Rosenborg has remained unscathed by war and fire alike. There are world-class examples of Renaissance craftsmanship, in particular Italian pietra dura inlay and intricate carvings in amber and ivory. Several small rooms lead off the expansive banqueting hall, one housing an exquisite Flora Danica collection (the finest Royal Copenhagen porcelain), another filled with Venetian glass of every colour. The most amusing room is mirrored throughout, including the floor and ceiling: guaranteed entertainment for the king in the days of ladies with hooped skirts and crinolines. (Oster Voldgade 4A, tel:  3315 3286).
7 Den Hirschsprungske Samling
This small museum tucked away in a park behind the overrated State Museum of Art is lesser known but well worth a visit. Most of the collection was put together by Heinrich Hirschsprung, a 19th-century tobacco manufacturer who bought paintings, drawings and sculptures from the artists of his time. He offered it to the state in 1902 on the understanding that an intimate museum would be specially built to house it. The country's most important artists of that period are represented here, as well as many more from the 20th century. Much of the furniture in the building comes from the artists' own homes, recreating the atmosphere of the era. Beneath one particularly lovely painting by Vilhelm Hammershoi, for example - an 1897 Louis XVI-style interior in soft pink tones - stands one of the very chairs that appear in the painting (Stockholmsgade 20, tel:  3342 0336; www.hirschsprung.dk).
8 Museum of Decorative
Art Wander along Bredgade and the string of antique and interior design shops will get you in the mood for the Museum of Decorative Art. Housed in a former 18th-century hospital, the Kunstindustrimuseet carries Denmark's largest collection of 20th-century craft and industrial design, and traces the history of European decorative art since the Middle Ages. The architects Arne Jacobsen and Verner Panton changed the face of Danish furniture design, and looking at their work may well inspire you to bring home something larger than expected. If you arrive in time for lunch, the food in the restaurant is good (Kunstindustrimuseet, Bredgade 68, tel:  3318 5656; www.kunstindustrimuseet.dk).
The image of Nyhavn harbour is often used to promote Copenhagen, and the cobblestones, colourful facades and wooden sailboats are picture perfect. No wonder Hans Christian Andersen chose to live in at least two of the buildings during his time in the city. The lustful sailors have long since abandoned this former red-light district, which turned from naughty to nice as late as the 1980s. Even so, a few tattoo parlours and go-go dancing clubs still give a flavour of its past. Pop into the Amber Museum to view a fascinating display of gems embedded with prehistoric insects and plantlife (Kongens Nytorv 2), and check out the shop for presents and jewellery. Wander into the side streets for antique shops, funky vintage kitchen equipment (Epoke, Lille Strandstraede 10) and Danish designer fashion (Bitte Kai Rand, Lille Strandstraede 22). If you feel you can't leave Denmark without at least a glimpse of the Little Mermaid, hop aboard a boat at the memorial anchor for a tour of the harbour and canal network (tel:  3342 3320). Edvard Eriksen's statue may not be the most exciting part of the trip, but the houseboats and historical buildings of Christianshavn - a kind of small-scale Amsterdam - make up for it.
Far more than just a funfair for children, Tivoli is a fairytale garden for all. While some are drawn to the big dippers, circus performers and giant ice creams, others go for concerts, a romantic row on the boating lake or a night at the theatre. Because the amusement park opened some 160 years ago, it holds an olde worlde charm far removed from the pop music and candy floss travelling fairgrounds most common in Europe. The atmosphere is defined by quaint restaurants, flowers and the delicious smell of freshly baked waffles, while at nightfall the place is set aglow with thousands of lanterns and fairy lights. There are excellent restaurants of every variety, as well as snack bars for those in search of the cheap and cheerful. (Fiske filet med remoulade - breaded fish served with a vegetable mayonnaise dressing - and smorrebrod or open sandwiches on rye bread are what the Danes do best.) Although Tivoli is open only from April to September, there is a wonderful Christmas market from mid-November to December 23, when you can ice-skate at the foot of the famous Chinese tower (Vesterbrogade 3, tel:  3315 1001; www.tivoli.dk).