Hamlet is such an often-quoted play, it is almost impossible to select just a few well-known quotes, but here are some likely contenders, in order of appearance in the play.
Included are some that are also often misquoted...
'O, that this too, too solid flesh would melt/Thaw, and resolve itself into a dew' (Hamlet, Act 1 Scene2)
'Neither a borrower nor a lender be / For loan oft loses both itself and friend / And borrowing dulls the edge of husbandry' (Polonius, Act 1 Scene 3)
'...though I am native here/and to the manner born; it is a custom / more honoured in the breach than the observance' (Hamlet, Act 1 Scene 4)
'Something is rotten in the state of Denmark (Marcellus, Act 1 Scene 4)
'That one may smile and smile and be a villain' (Hamlet, Act 1 Scene 5)
'There are more things in heaven and earth, Horatio/Than are dreamt of in our philosophy' (Hamlet, Act 1 Scene 5)
'Though this be madness, yet there's method in't' (Polonius, Act 2 Scene 2)
(This is often summarised or misquoted as: 'There's method in his madness')
'O, what a rogue and peasant slave am I!' (Hamlet, Act 2 Scene 2)
'To be or not to be, that is the question' (Hamlet, Act 3 Scene 1)
'The lady protests too much, methinks' (Gertrude, Act 3 Scene 2)
(This is often misquoted as: 'Methinks the lady doth protest too much')
'How all occasions do inform against me, and spur my dull revenge'
(This famous soliloquy by Hamlet appears only in the Second Quarto in an extended Act 4 Scene 3)
'Alas, poor Yorick, I knew him, Horatio. A fellow of infinite jest' (Hamlet, Act 5 Scene 1)
(This is often misquoted as: 'Alas, poor Yorick, I knew him well')
'If it be now, 'tis not to come: if it be not to come, it will be now: if it be not now, yet it will come: the readiness is all.' (Hamlet, Act 5 Scene 2)
'The rest is silence' (Hamlet, Act 5 Scene 2)
Written by Kathleen Bradley © RSC
Photo shows Kenneth Brannagh as Hamlet in the RSC's 1992 production © RSC/SBT