What is the smallest and fastest possible query in MySQL? What would be the fastest possible query to run in MySQL, that will allow monitoring of the basic service functioning?

A few examples I can think of:

SELECT NOW();


or 

SHOW GLOBAL STATUS LIKE 'Uptime';


etc..

The purpose is to execute the SQL every few minutes, so the performance is key.
@Kondybas is correct; Fastest query is SELECT 1.

@a-horse-with-no-name is also correct "If your server can't cope with a statement like that once a minute, something is seriously wrong."

Percona Monitoring and Management monitors MySQL servers, in some cases, once per second, without "performance issues".