（スーパーなどの）チラシって英語でなんて言うの？

A paper with advertisements is called a flyer and then a paper with advertisements that is folded is called a brochure

All the answers refer to sheets of paper which are handed out free of charge.
These sheets of paper are used to advertise events, products or businesses.

Note that a flyer and a leaflet are usually smaller in size.
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Example
A: Help me
B: With what?
A: Hand out flyers at the gym.

A: What is that?
B: A gym pamphlet

A) pamphlet -a small booklet or leaflet containing information or arguments about a single subject.
Example -"he published a spate of pamphlets on the subjects about which he felt strongly"

B) Flyer -a small handbill advertising an event or product.
Example -"he was handing out flyers promoting a new sandwich bar"

I hope this helps :-)

You may be handed a flier in tyhe street withdetails of a newly-opened restaurant, for example. A brochure would normally be an information sheet detailing a company's products, and this may also be handed out in the street or at exhibitions, or distributed to peoples homes by way of a 'leaflet drop.'

We call the single sheet advertisements from a store, etc. a 'flier' (flyer).

Mailshot = the posting of an advertisement so that it reaches a number of people at one shot.
An ad (short for advertisement)

A paper with advertisements written on it, are most commonly referred to as an "ad" or a "flyer".

Example sentences: I was shopping in the city and I was given a flyer advertising for a new local gym that is opening.

I was given an ad whilst I was out and about today.