The worldwide stage and cinema hit Mamma Mia! became the biggest-selling DVD of all time in the UK, and its director Phyllida Lloyd is appointed a CBE.
She said the honour was a "complete surprise", and had left her feeling "humbled".
Paying tribute to her colleagues, she added: "One gets to be the captain of the ship, but ultimately you would not get anywhere without the cast and crew."
Button said that being appointed an MBE was the "crowning glory" of a year that had begun with him unsure of his involvement in Formula One after his car-maker, Honda, quit the sport.
Button secured the 2009 drivers' title in the penultimate grand prix in Brazil
"The past 12 months have provided me with such an exhilarating journey," he said. "We literally started with nothing, but had the belief, determination and passion to take on the world and succeed."
Button's boss, the man who led a management buy-out and took the new Brawn GP team to the constructors' title, 55-year-old Ross Brawn, becomes an OBE.
"Everybody at Brawn GP played their part - and it was down to their expertise, dedication and hard work that we were able to achieve so much," Button, 29, added.
Sir Ian McGeechan is honoured for services to rugby in a distinguished playing and coaching career at club level, for Scotland and the British Lions.
He was coach of the 1990 Gland Slam-winning Scotland side, and has masterminded two Lions Test series wins in Australia in 1989 and South Africa in 1997.
Sir Ian played on two British Lions tours, later coaching on five trips
Dedicating the knighthood to his family, 63-year-old Sir Ian said: "It is their constant support that has provided such an inspiration to me and this honour would not have been possible without them."
Among other sports stars honoured is Britain's greatest ever gymnast, the two-time world champion Beth Tweddle.
She is appointed an MBE after last year's remarkable gold medal win at the World Championships in London, where she triumphed in the less-favoured floor competition following a fall in her signature event on bars.
Tweddle, 24, said the honour had come as a "total shock", adding: "I spoke to my grandparents at the weekend because I didn't want them to find out through the newspapers first, and they just broke down in tears.
"That just showed me how much it meant to them, and how much these awards mean to people in general."
Cricketer Claire Taylor is made an MBE after a year in which she helped England to the World Cup and World Twenty20 titles.
She was named player of the tournament at both events and Women's Cricketer of the Year by the International Cricket Council, having become the first woman to make Wisden's Five Cricketers of the Year list earlier in 2009.
Some 979 people are recognised in the New Year Honours.
There is a damehood for Dr Claire Bertschinger, whose appearance in BBC News reports by Michael Buerk of the 1984 Ethiopian famine helped inspire a global aid effort.
Her account of having to choose which starving children to feed galvanised Bob Geldof to organise the Band Aid recording and Live Aid concert.
Leaving school at 15 years old, never would I have dreamed that I would be recognised in this way
Ultimo boss Michelle Mone OBE
Of her honour for services to nursing and international humanitarian aid, Dame Claire said: "It's an enormous honour. I was very moved and startled by receiving it."
There is a knighthood for Erich Reich, chairman of the Kindertransport Group of the Association of Jewish Refugees, who has helped raise millions of pounds for charity.
Sir Eric was four when he arrived in the UK as one of 10,000 "Kindertransport" children evacuated from Nazi-occupied Europe.
Fashion industry figures are honoured with Ultimo lingerie boss Michelle Mone, designer Amanda Wakeley and milliner Stephen Jones all appointed OBEs.
Mone, who has appeared as a contributor on BBC's The Apprentice: You're Fired show, said: "Leaving school at 15 years old, never would I have dreamed that I would be recognised in this way."
Designers Luella Bartley and Cath Kidston, and London-based tailors Timothy Everest and Imtaz Khaliq are appointed MBEs.
Children's author and illustrator Lauren Child, creator of the Charlie and Lola series, is awarded an MBE and Anthea Bell, co-translator of the Asterix comic books, gets an OBE.
Rose Gray and Ruth Rogers, the co-founders and chefs of the Michelin-starred River Cafe restaurant in west London, are also appointed MBEs.
There is a first ever honour for a circus ringmaster, with 73-year-old Norman Barrett of Zippo's Circus becoming an MBE.
"This is a real boost for circus and in my view gives the art form the recognition it truly deserves and, sadly in modern times, so rarely gets," he said.
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