It was a disappointing session for Woods, who carded three bogeys with some ragged play on a day when he was aiming to get in position to end a four-year title drought in golf's four Major tournaments.
The former world number was one-under for the tournament and five strokes behind McIlroy and Singh after seven holes when a siren sounded to call players off the course. Play is scheduled to resume at 7:45 a.m. local time on Sunday.
"I got off to a rough start today and couldn't get anything going. I'll come back tomorrow morning and see what happens," said Woods. "There are a lot of holes left to play."
While Woods was relieved that rainy weather cut short his troubles and gave him a chance to start fresh on Sunday, McIlroy had to interrupt a round where he was starting to hone in on a second Major title.
Typically though, the Northern Irishman, who faces 27 holes now on what will be an extended and gruelling day on the longest course ever used for a Major, took the setback in his stride.
"It's fine, I don't think you can let it affect you at all. The way I'm looking at it, I'm going into the final day of the final Major of the season tied for the lead, so I can't ask for much more," said McIlroy.
"I don't care if it is going to be 27 holes, 18 holes, 36 holes, I'm just happy to be in a good position."
After a threat of a lightning caused officials to bring the players off the course, heavy rain followed, forcing 26 players to finish their third rounds on Sunday.
McIlroy, playing some sparkling golf, was four-under after nine holes on Saturday and six-under for the tournament while Singh has 11 left to play and Woods has eight holes remaining.
The confident 23-year-old had five birdies in his opening eight holes for the outright lead before a bogey at the ninth.
McIlroy even came through a bizarre situation on the third hole where his tee shot hit a tree and got lodged in the thick of a branch, leaving him and his caddy searching in vain for the ball on the ground until a member of a TV crew pointed out its location to them.
Woods undid much of the good work he did in Friday's classy 71 as the 14-times Major winner missed a four-foot putt for birdie on the third but then paid the price more for loose shots from the tee.
On the par-four fourth Woods drove wide left and then found a difficult lie with his second shot, failing to reach the green but he at least restricted the damage to a bogey with a decent up-and-down.
On the par-three fifth, Woods landed short of the green and then missed a 10-foot par putt while on the par-five seventh he twice found sandy areas and missed a par putt from 15 feet.
Another of the trio of overnight leaders, Singh, US PGA winner in 1998 and 2004, started off in perfect fashion with a 14-foot birdie putt on the first and a 20-foot putt on the seventh that put him at six-under and level with McIlroy.
Australia's Adam Scott, who narrowly lost The Open to Ernie Els last month, was bogey free and made four birdies to make 32 in his front nine, leaving him just a shot off the pace.
Sweden's Carl Pettersson, the other member of the trio of second round leaders, recovered from his bogey on the third with a birdie on the par-three fifth to remain at four-under.
Three Americans put themselves in the frame for Sunday's final round as Bo Van Pelt, Steve Stricker and Jimmy Walker all carded five-under 67s.
Van Pelt held the clubhouse lead at three-under, while Stricker was a further shot back and Walker was one-under.
South Africa's Trevor Immelman, who has not had a top-10 finish in a Major since winning the Masters in 2008, was level with Van Pelt after making three birdies and a bogey from 10 holes.