The Golden Star Warriors had almost 70% of possession and twenty shots per goal but could not find a way to pierce Singapore's defence.
Korean coach Park Hang-seo made eight changes to the starting line-up from the 3-0 win against Malaysia on December 27 in Hanoi.
Veterans Dang Van Lam, Que Ngoc Hai and captain Do Hung Dung stayed in the starting 11.
On the other side of the pitch, Singapore's Takayuki Nishigaya started Hassan Sunny in goal, replacing Hafiz Nor, and Shawal Anuar with Ryhan Stewart and Irfan Fandi in defence.
The first dangerous chance of the match came to Vietnam in the 11th minute when Nguyen Van Quyet's shot forced Hassan Sunny to parry. The rebound reached Pham Tuan Hai, but his shot went straight at Sunny.
Sunny was troubled again six minutes later when Vu Van Thanh fired a vicious shot outside the box, but the Singaporean keeper made a simple save.
Deep in their half, Singapore looked for opportunities on the counter. In the 26th minute, from Shakir Hamzah's left-wing cross, Ilhan Fandi tried a shot from outside the box, but the ball went wide.
Vietnam replied just minutes later when Van Quyet seemed to have capitalised on a Singaporean defender's mistake. From 25-metres, the Hanoi FC captain's shot went way over the bar.
Just before the break, Van Thanh was freed in the right wing by Hung Dung and crossed inside the box. The powerful cross hit a Singapore defender and rebounded to the side netting.
Heavy firepower was brought on by Park after the break, in the form of superstars Nguyen Quang Hai and Phan Van Duc, who replaced a fatigued Hung Dung and a dismal Chau Ngoc Quang.
The changes, however, did not work immediately. Nguyen Hoang Duc and Nguyen Tien Linh were brought on in the 58th minute.
Just four minutes after Duc and Linh were subbed on, the two combined and found Sunny's net. However, the goal did not stand because the referee paused the match after Van Thanh was brought down.
Seconds later, Vietnam won a free kick near the six-yard box. But from his favourite position, Quang Hai tried a lobbed shot that went just wide.
Tien Linh, who Ngoc Hai jokingly compared to Argentina's Lautaro Martinez for his missed opportunities, had around five clear chances after he was subbed on.
In the 65th minute, Linh could have won it with a belter for Vietnam, but his volley outside the box was bravely saved by Hassan Sunny, who went down injured later.
Ten minutes later, it was Linh again who had time and space on his side, but the striker's shot was too rushed and not on target.
Luck was not on Vietnam's side again in the 84th minute, when Hoang Duc's 30-metre shot beat Hassan Sunny, but the effort hit the post to the agony of the Vietnamese supporters at Jalan Besar Stadium.
Vietnam had two more chances to win the game, but Tien Linh squandered both and the match finished 0-0.
With this result, Vietnam's position at the top of Group B still stands, but they are not yet qualified for the knockout stage.
In the unlikely event of the Golden Star Warriors losing by seven goals at home to already-eliminated Myanmar on January 3, as well as a draw in the match between Singapore and Malaysia happens, Park's team will be eliminated.
However, a win is needed for Vietnam to avoid trickier opponents in the knockout stage, as well as rebuild confidence for the squad.
