What are some examples of syllogisms?

The most famous one is "All humans are mortal. Socrates is human. Hence Socrates is mortal" Which (by the way) he proved by committing suicide.

As obvious as it may seem in its simple structure, the Syllogism is a powerful Logical instrument. You start with a general statement, you follow up with an observation and you come to the conclusion. The modern scientific reasoning does little more than this.

BEWARE : syllogism must be used from general to particular, never the other way. You could not say "Socrates is mortal. All humans are mortal, All humans are Socrates".

In the history of Science examples of inverse syllogisms do exist, but in that case it in not logic, it is serendipity.