On someone's trail phrase
The expression can be rewritten as "on the trail of someone".
If a person is "on his/her trail", he/she is following, tracking someone in order to find them or find out about them.
He must have done something wrong or even illegally. That's why the police in on his trail.
A: "What are you doing here?" B: "I'm on that girl's trail. I have found something abnormal from her computer. She's a suspect in a case".
1. To punch a hole in one's ticket to show that they have paid for a ride
2. To do something that allows one to go further in a competition or to be promoted at work
1. Please, tell me if I forget to punch your ticket.
2. With a view to punching my ticket to the final, I did whatever it takes to win this round.