Singh and Bourdy shrugged off the rainy, windy conditions to shoot matching seven-under 65s as the tournament returned to Northern Ireland for the first time since 1953.
Britons Mark Foster and Andrew Marshall were in joint third place on 66 alongside Frenchman Edouard Dubois, Finn Mikko Ilonen, Swede Oscar Floren and Australia's Matthew Zions.
Triple major champion Padraig Harrington was among a group of players on 67. Former US Open winner Graeme McDowell and Open champion Darren Clarke carded 71s.
The appearance of Northern Irish trio McIlroy, McDowell and Clarke and Irishman Harrington at the Royal Portrush links course in County Antrim has helped the European Tour to achieve a first by declaring a sellout for all four days.
"The atmosphere and the feel to the golf course is fantastic," said the 40-year-old Singh. "When you have so many people cheering and watching you I think you feel great.
"I love links golf. You have to hit a lot of low shots and you have to have a lot of imagination."
Englishman Foster said fans had been waiting to get on the course at six o'clock in the morning.
"There were people queuing to get in even though the rain was coming sideways," he said.
"I've never seen a buzz in the players' lounge like there is this week. People are raving about the course and the size of the crowds."
McIlroy, watched by girlfriend Caroline Wozniacki following her first-round defeat at the Wimbledon tennis, was four-under for his round until he three-putted the 16th and 18th to finish with a 70.
"As the day went on the greens got a bit slower and coming from those I've been putting on in the United States it's a big change," he said.
McIlroy has missed the cut in four of his last five outings.
US PGA champion Keegan Bradley struggled to a 73 in his first competitive round in Europe.