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Cavendish, who has won 30 stages at the Tour de France, came sixth in the first of six events - the scratch race.
He was second fastest in the individual pursuit and seventh in the elimination race with three more events on Monday.
Becky James broke the women's sprint Olympic record in qualification and is in Monday's quarter-finals along with fellow Briton Katy Marchant.
Cavendish, 31, began his campaign with a hard-fought sixth-placed finish in the scratch race, after Denmark's Lasse Norman Hansen, the London 2012 Olympic champion and Roger Kluge took the top two spots by gaining a lap on the field.
In the individual pursuit, Cavendish recorded a time of four minutes 16.878 seconds and caught reigning world champion Colombian Fernando Gaviria Rendon in the process.
That temporarily gave the Manxman the outright lead, before Hansen set a new Olympic record of 4:14:982, beating Sir Bradley Wiggins' time set in 2008, to win his second event in a row.
However, Hansen dropped from first to sixth overall after he was the first rider to fall out of the elimination race which sees the rider in last position at the end of every second lap ejected from the race.
Cavendish was looking comfortable as other riders dropped out but he was ejected, despite not being the last rider over the line, when he overtook a rider while off the inside of the track, and was awarded seventh place.
France's Thomas Boudat, the 2014 world champion, leads on 106 points, two more than Italy's Elia Viviani who won the elimination race, with Cavendish third on 96 points.
The fourth event - the time trial - starts at 14:21 BST on Monday, before the flying lap and points race events.
Wales' James won a silver medal in the keirin on Saturday and maintained her good form on Sunday as she set a new Olympic record during the sprint qualification.
James recorded a time of 10.721 seconds to beat the 10.724 set by fellow Briton Victoria Pendleton at the 2012 Olympics in London.
That saw James become the fastest qualifier and she comfortably beat Olga Ismayilova of Azerbaijan by a margin of 0.165 secs to move into the quarter-finals, which take place from 14:00 BST on Monday.
Marchant also advanced in style as she defeated Canada's Monique Sullivan after posting the second fastest qualification time.
Australia's defending champion Anna Meares was beaten by Lithuania's Simona Krupeckaite but won a repechage race to reach the last eight.
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Great Britain's Mark Cavendish is third after day one of the men's omnium as he aims for his first Olympic Games medal.