Why go now?
The Danish capital is a city on a human scale, ideal for strolling and sightseeing even in freezing-cold January. Copenhagen warms up with the 10-day Winter Jazz Festival ( jazz.dk) that starts this Friday, 28 January, at more than 50 venues across Denmark and southern Sweden, including the recently reopened Jazzhus Montmartre (1) at Store Regnegade 19A (00 45 33 32 96 66; jazzhusmontmartre.dk) and the Royal Danish Theatre (2) at Kongens Nytorv 9 (00 45 33 69 69 69; kglteater.dk). Getting there is cheap if you travel before Easter and avoid the Valentine's Day crowds.
I paid £55 return with easyJet (0905 821 0905; easyjet.com) which flies from Gatwick, Stansted and Manchester. British Airways (0844 493 0787; ba.com) and SAS (0906 294 2772; flysas.com) fly from Heathrow, with SAS and its partner BMI (0870 60 70 555; flybmi.com) flying also from Aberdeen, Birmingham, Glasgow and Manchester.
Norwegian (020-8099 7254; norwegian.com) flies from Gatwick and Edinburgh, CimberAir (00 45 70 10 12 18; cimber.com) from Gatwick and Newcastle and Air Berlin (0871 5000 737; airberlin.com) from Stansted.
Copenhagen's stylish Kastrup airport is only five miles south of the city centre; the 13-minute trip to the Central Station (3) costs 34 Danish kronor (Dkr34/£4) return.
Get your bearings
Copenhagen lies on the eastern side of the Danish island of Zealand (Sjælland in Danish). The city centre is framed by Amsterdam-style canals which intersect splendid historic buildings and avant-garde architecture. The Old Town is small enough to walk through in around 25 minutes.
The historic heart of the city is the island of Slotsholmen, home to the Danish Parliament and a bevy of state museums. Nearby, Indre By is the commercial centre of Copenhagen, where most of the restaurants and shops are located.
Christianshavn, to the south, hosts the alternative "Free City" of Christiania. Further out are multicultural Vesterbro, leafy Frederiksberg and fashionable Nørrebro.
Lovers of mid-century style should visit the birthplace of the original designer hotel. The Radisson Blu Royal Hotel (4) at Hammerichsgade 1 (00 45 33 42 60 00; radissonblu.com) is a modernist skyscraper designed by the revered Danish architect Arne Jacobsen.
With its grey, blue-green colours, wengé wood and late-Fifties furniture, this hotel feels like a set from the American drama Mad Men. All 260 rooms have good views, particularly those overlooking the Tivoli Amusement Park (closed until 14 April). Room 606 is the only room with the original Arne Jacobsen décor. Doubles start at Dkr2,500 (£305), room only.
In the alluring Nyhavn area, try the Hotel Bethel Somandshjem (5) at Nyhavn 22 (00 45 33 13 03 70), a former seamen's hostel that has been elegantly converted; doubles from 795Dkr (£90) including breakfast.
The trendy Danhostel (6) at Hans Christian Andersens Boulevard 50 (00 45 33 11 85 85; danhostel.dk) is the largest city-centre hostel in Europe, with more than 1,000 beds – and some great views over the harbour. A twin room with en-suite bathroom costs Dkr499 (£55), including breakfast.
Take a hike
Starting at Town Hall Square (7), stroll down Strøget. Weave in and out of the little side streets to find a selection of hidden cafés and other gems. You'll arrive at the city's central square, Kongens Nytorv (King's Square), where you can admire its grandest occupant: the late 19th-century Det Kongelige Teater (8). Now get your camera ready to capture Copenhagen's favourite postcard view as you walk across to Nyhavn (New Harbour). Built in 1671 to bring ships to the city centre, this picturesque canal was once a by-word for debauchery as the main red-light district.
Nowadays, bars and restaurants occupy the multi-coloured former merchant's homes. Tables and chairs spill out onto the cobbled promenade and attract al-fresco diners (mainly Norwegians) even in sub-zero temperatures.
Take a ride
If the cold is starting to bite, it's a good idea to jump aboard a DFDS City Canal tour (00 45 32 96 30 00; canaltours.dk) from the pier (9) at Nyhavn. Heated boats leave every hour from 10am to 3pm, price Dkr70 (£7.80). The Grand Tour route allows the city's delights to unfold before your eyes including the Queen of Denmark's stately (but a little lacking on opulence) palace, Amalienborg Slot (10), before heading into the narrow canals of the offbeat Christianshavn neighbourhood – nicknamed "Little Amsterdam".
Lunch on the run
Emulate the locals and get a sugar hit at the wonderful canalside branch of Lagkagehuset (11) at Torvegade 45 in Christianshavn, with shelves of home-made pastries, muffins and bread. Try the Christianshavner Tarte (Dkr26.50/£3), a delicious mix of biscuit and nut base with a caramel, strawberry and cream topping.
Danish design greets you everywhere, from chic boutiques through to furniture stores. Shops close early on Saturdays and most are closed on Sundays. The impressive silversmith Georg Jensen (12) at Amagertorv 4 in Indre By (00 45 33 14 02 29; georgjensen.com) showcases stylish items from cutlery, jewellery and candlesticks. Bargain hunters can pick up gems in the various bric-a-brac shops and fleamarkets in Nørrebro, particularly in and around Ravnsborggade (13), which is usually open between 10am and 4pm.
Learn more about the city's history at the København Bymuseum (14) (City Museum), at Vesterbrogade 59 (00 45 33 21 07 72; copenhagen.dk). A new interactive exhibition (open until December 2012) explores the sometimes controversial legacy of recent immigration, from Jewish exiles from Nazi-occupied Europe to more recent Iraqis and Somalis. Open 10am-4pm daily, admission Dkr20 (£2.30).
Whether you want to contemplate the future of furniture or just get some inspiration for the spare room, an hour or so at the Dansk Design Centre (15) at Hans Christian Andersens Boulevard 27 (00 33 69 33 69; ddc.dk) is a worthwhile stop. The new exhibition "Denmark by Design" (open until 2013) chronicles the development in Danish design from 1945-2010. It is open 11am-4pm at weekends, 10am-5pm for the rest of the week (to 9pm on Wednesdays), admission Dkr55 (£6.15).
The Dutch-inspired Café Quote (16) at Kongens Nytorv 16 (00 45 33 32 51 51; cafequote.dk) has a wine list as long as Stroget. Enjoy a glass of Pinot Noir (Dkr52/£5.80) overlooking King's Square.
Dining with the locals
Copenhagen is riding the crest of a culinary wave after the mercurial René Redzepi's Noma (17) at Strandgade 93 (00 45 32 96 32 97; noma.dk) was voted best restaurant in the world by a survey of international food critics and chefs last year.
But unless you book months in advance, seek out other Danish-French fusion restaurants.
L'education Nationale (18) at Larsbjørnsstræde 12 (00 45 33 91 53 60; leducationnationale.dk) is a bistro which serves hearty French classics such as rabbit casserole with main courses costing around Dkr200 (£23).
CityKroen (19) at Gammel Torv 8 (00 45 33 91 13 30; citykroen.dk) offers a cosy setting and meat dishes such as wild boar with cranberries; mains average Dkr149 (£17).
Sunday morning: out to brunch
O's American Breakfast & Dinner (20) at Gothersgade 15 (00 45 33 12 96 12; osamerican.dk) is a haven for the hungover who flock here to ease their sore heads by enjoying the house speciality of pancakes with bacon and bottomless cups of coffee; Dkr79 (£9).
Go to church
Vor Frelsers Kirke (21) (the Church of Our Saviour) at Sankt Annægade 29 (00 45 32 57 29 98; vonfrelserskirke.dk) dominates the city's skyline. Completed in 1696 in Dutch Baroque style, the church has an impressive altar and a huge organ resting on top of two stucco elephants. But the real reason to visit is the lavish 300-foot spire, which you can climb in 400 steps – the last 150 of them on the outside, on a golden staircase that coils around the tower like a snake. Local legend insists King Christian rode his horse all the way to the top in 1752. The views are still fantastic. Open noon to 3.30pm.
A walk in the park
Escape to Copenhagen's "city within a city" in the leafy, villa lined suburb of Fredericksberg. Its huge, pleasant park (Frederiksberg Have) is a perfect break from the crowds and is peppered with a host of interesting sights. The English-style gardens are criss-crossed with pathways, lakes and canals which make for a romantic stroll in the bleak winter sunshine. Then admire the grand summer palace, Fredericksberg Slot (22), particularly in dappled light, when its Italianate-style 18th-century yellow façade looks stunning.
On the south side is the rather eccentric Cisternerne: Museet for Moderne Glaskunst, which is a museum of glass housed underground in a former water cistern. It opens only from the start of February, and then only at weekends: 11am-5pm; Dkr50 (£5.70).
The icing on the cake
With a cup of steaming Gløgg in hand, wander along to Copenhagen's most celebrated landmark – the Little Mermaid (23) on the shore at Langeline – and engage in some fascinating people-watching. This rather diminutive and ordinary bronze statue was inspired by the Hans Christian Andersen story of the same name. She is arguably the world's most photographed woman – and has only recently returned from her own holiday after spending much of last year on display at the Danish exhibition at the Shanghai Expo last year.
But stand well back and marvel at the true spectacle on show, namely the hordes of tourists who clamber over the rocks and strike no end of silly poses.
Additional research by Libby Galvin