Can lest be used without the support of the word should? Yes. And it normally is used without should. In Shazad's example, 'lest you should fail your examination', that use of should of course has a completely different meaning from the usual meaning of should.

Is lest always followed by should?

The correct sentence should be: Study hard lest you should fail. ** Lest is always followed by should, with a 'not' anywhere. 'should' here can not be replaced by 'would' or 'may'.

What is the rule of Lest?

/lest/ (formal or literary) ​in order to prevent something from happening. He gripped his brother's arm lest he be trampled by the mob. Lest anyone should doubt my story, I have brought documents to attest to its truth.