On the up-and-up on the up and up
To be honest/trustworthy/sincere.I do not like that company because they are not on the up-and-up.
On the trail/track of (someone or something)
Seeking someone or something.The police dogs were on the trail of the bank robber.
On the tip of one`s tongue
Not quite able to remember something.The name of the movie is on the tip of my tongue.
On the take
Accepting bribes.The border guard was discovered to be on the take and was immediately fired.
On the strength of (something)
Due to something such as a promise or evidence.On the strength of my high test scores I was admitted to the university that I wanted.
On the spur of the moment
On a sudden wish or decision, suddenly, without planning, on impulse.I bought a new bicycle on the spur of the moment.
On the spot
In a difficult or embarrassing situation.The man was suddenly put on the spot when the reporter asked him about the money.
On the sly
Secretly.We went to the restaurant on the sly so that nobody would know where we were.
On the same wavelength
Thinking similarly about something.Everybody is on the same wavelength about the need for change in the company.
On the wrong track
Following the wrong set of assumptions, going the wrong way.The police were on the wrong track when they went to question the owner of the store.