Where & why 31 and 32 subnet mask are used?

one example I can think of currently
ISP <-->R1 --(your network)
as per addressing rules, the segment between ISP and R1 need to be in the SAME network. Many times this will be a serial link and due to which you need only 2 addresses for each side. which will be a /30 network address. (2^2 -2 =2 usable addresses)
but again there are few theories/RFCs. For point to point ( manily used in frame relay or Ppp) we 'dont' really end up broadcasting. 😀 as we have only 2 endpoints, hence it doesnot make a point to have a reserved 'broadcast' address. think of it like a water pipe moving water from tap to a drum. you wont see water spilling everywhere, it only flows in a gien channel. same is the case with point to point. You dont need a broadcast address. hence /31 address can also be used giving scope to 2 host address ,(2^1) - ie. host bit '0' represents one host and '1' represents another.
/32 is used to represent a host as it is.

Thanks Durga for your reply.

Still I have one question , in case of 3g modems( in all dial up) /32 is used, then how it communicate to server?

I an not too familiar with dailup configuration, but have another example for you for /32 addresses
you must have heard of classical loopback interfaces . right? Those can be treated as single /32 address.
Loopback interfaces generally are used to test the tcp stack within a given host , like - 127.0.0.1/32