Against the backdrop of accusations that Red Bull has broken Formula One's salary cap - claims, if proven correct, that will bring into question the legitimacy of Verstappen's title triumph over Lewis Hamilton 10 months ago - the Dutch driver qualified only eighth.
Ferrari's Charles Leclerc took pole position ahead of Verstappen's team-mate Sergio Perez. Hamilton finished third, missing out on his first pole of the season by just 0.054 seconds in some thrilling wet-dry qualifying at the Marina Bay Circuit.
Verstappen appeared on course to topple Leclerc with his final throw of the dice, only to be told to abort his lap by Red Bull because he would have been short of the required one litre of fuel.
Verstappen, in the dark over Red Bull's costly error, turned the airwaves blue.
"What the f***?" he yelled over the radio. "What the f***? What the f*** are you guys saying? It is unbelievable, mate. I don't get it. What the f*** is this about?"
Verstappen will take his second title in as many years if he wins on Sunday, and sets the fastest lap, and team-mate Perez and Ferrari's Leclerc finish lower than fourth and eighth respectively.
But with Leclerc and Perez sharing the front row, and Verstappen way down the order, the 25-year-old's hopes of securing the championship with five races to spare appear improbable.
"It's just incredibly frustrating and shouldn't happen," said Verstappen. "I am not happy at all at the moment.
"I know it's always a team effort, and I can make mistakes and the team can make mistakes, but it's never acceptable. Of course, you learn from it, but this is really bad for us."
For more than a moment, it seemed as though Hamilton, a four-time winner in Singapore, might take his first pole since last year's penultimate round in Saudi Arabia only for Leclerc and Perez to usurp him in the closing stages.
"I was pushing so hard, and it was so, so close," said Hamilton, 37. "I thought we could be fighting for first place but I didn't have grip on that last lap.
"I am grateful to be on the second row and the team for continuing to push. We keep our head down and hopefully tomorrow will be a better day. We get up and fight again."
Carlos Sainz qualified fourth for Ferrari, one place ahead of Fernando Alonso. Lando Norris was sixth for McLaren.
It was a Saturday to forget for George Russell after he was eliminated from Q2. The Englishman has enjoyed an impressive opening season as Hamilton's team-mate at Mercedes, finishing in the top five at every race he has completed. But Russell will struggle to keep that fine record intact after qualifying only 11th.
Alex Albon is back in the cockpit this weekend - three weeks after he suffered respiratory failure and was left on a ventilator following complications from surgery.
The London-born driver finished ahead of his soon-to-be-departing Williams team-mate Nicholas Latifi, but was only 19th of 20.