O'Shea was third overnight after finishing third in the flying lap and the elimination race while ending eighth in the points race.
A silver medalist in the team pursuit with Australia, O'Shea took command of the overall standings after posting the third best time in the individual pursuit, in 4 minutes 24.811 seconds.
Lasse Norman Hansen of Danemark had the best time, in 4:20.674, ahead of Edward Clancy of Britain, who was second in 4:20.853.
With the 15km scratch race and the 1-kilometer time-trial still to be raced on Sunday, O'Shea leads the standings with 17 points, with Clancy two points behind.
Clancy, who is bidding for a second gold medal on home soil after winning the team pursuit, was fourth following the first day of competition, which featured his two less favorite events.
Thanks to his comfortable win in the flying lap - a strong indicator of his fine shape - the 2010 world champion trailed overnight leader Bryan Coquard of France by only seven points after limiting damages in the highly tactical points and elimination races.
Focusing on the team pursuit this season, Clancy did not prepare for those events and finished 11th in the points race after winning three sprints but recovered by claiming the fifth place in the elimination.
The omnium is like a decathlon of cycling, apart from the difference that there is six events in the omnium. Mixing endurance and sprint races, the event is very popular in cycling schools because it offers youngsters a chance to develop their versatility.
Each rider receives a score in each of the six races equivalent to their finishing position. The rider with the lowest score wins the competition.
The UCI only announced its decision to add the event to the Olympics in 2010, giving only two years to nations to prepare for it.
Coquard moved down to fifth overall after finishing 12th in the pursuit. Hansen was third overall with 19 points, one less than Italian Elia Viviani.
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