Masters legends Nicklaus and Player got the 82nd Masters off to a rousing start by hitting the ceremonial tee shot before turning the first tee over to American Austin Cook, who was the first of the field of 87 golfers to tee off.
Woods, who said it was only after playing an exhibition last December that he felt he would be able to compete in the year's first major, has since shown flashes of the form that helped him earn a staggering 79 PGA Tour wins.
Player and Nicklaus continued their role as honorary starters by each hitting a tee shot on the opening hole, with 82-year-old Player producing the longest - and most accurate - drive by around five yards from 78-year-old Nicklaus.
Singh, 55, who won at Augusta National in 2000 as Woods finished fifth, birdied the first, second and fourth to lead by two shots from 61-year-old Mark O'Meara, the champion in 1998.
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Leishman missed good birdie chances on 11 and 12 but remained at the top of the leaderboard on three under, while Rory McIlroy had birdied the opening hole for the first time at the 35th attempt.
Woods may have won the last of his four Green Jackets in 2005 and not claimed a major victory since the 2008 U.S. Open but the former world number one will have the backing of massive galleries when he tees off at 10:42 a.m. ET (1432 GMT) with Australian Marc Leishman and Englishman Tommy Fleetwood.
Spain's Sergio Garcia, who ended almost two decades of major futility with his victory here previous year, played in the group behind Woods and opened with a par in his bid to join Nicklaus, Nick Faldo and Woods as the only players to successfully defend the Masters crown.
So was Northern Ireland's Rory McIlroy, who can complete a career Grand Slam with a victory this week.
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In a talent-laden field, 47-year-old five-time major victor Phil Mickelson, world number one Dustin Johnson and former Masters victor Jordan Spieth were to tee off later on Thursday.
Johnson, who has reigned at number one for more than a year, has catching up to do after a pre-tournament slip on the stairs forced him out of last year's Masters at the 11th hour. "I mean, I have played here and I've won here not playing my absolute best, but there's got to be a certain part of my game that's on", said Woods.
He has produced plenty of spectacular moments at the Masters - starting with his record-setting first major triumph at the age of 21 in 1997.
That 68 in 2010 - when he was returning to action after a humiliating sex scandal that wrecked his marriage and sent sponsors fleeing - proved the springboard only to a tie for fourth.
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If Woods wins the 2018 US Masters this week, Faldo says it would be "the greatest comeback in sport".