Looking on the internet deeply has found these results:

on the swing is the most popular phrase on the web.

in the swing 112,200,000 results on the web	on the swing 114,600,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
mail, Karl. Got your E It's to get you back in the swing. Papa's in the swing Takes a little time to get back in the swing of things. If you put her in the swing she'll get sleepy. You're keen to get back in the swing, and that's to be commended... But now I'm in the swing of it again.	And I was on the swing and then all of a sudden my hair got tangled in the chain. I want to go and play on the swing. Okay, get on the swing. I'm going out on the swing to read. Dennis, I told you to stay on the swing! Dennis! Daddy, I want a turn on the swing set. No. I was just sitting on the swing in my pajamas and swinging, enjoying that smell and that water and that moisture in the air. You have reached your maximum potential energy on the swing.

in the swing

112,200,000 results on the web

on the swing

114,600,000 results on the web

More popular!

And I was on the swing and then all of a sudden my hair got tangled in the chain.
I want to go and play on the swing.
Okay, get on the swing.
I'm going out on the swing to read.
Dennis, I told you to stay on the swing! Dennis!
Daddy, I want a turn on the swing set. No.
I was just sitting on the swing in my pajamas and swinging, enjoying that smell and that water and that moisture in the air.
You have reached your maximum potential energy on the swing.