I mean, have a look at this list. It's not even nearly complete, I took out a bunch of less common, but still very recognizable phrases.
All that glitters is not gold (The Merchant of Venice)("glisters")
All's well that ends well (title)
As good luck would have it (The Merry Wives of Windsor)
As merry as the day is long (Much Ado About Nothing / King John)
Bated breath (The Merchant of Venice)
a charmed life (Macbeth)
Be-all and the end-all (Macbeth)
The better part of valor is discretion (I Henry IV; possibly already a known saying)
Neither a borrower nor a lender be (Hamlet)
Brave new world (The Tempest)
Break the ice (The Taming of the Shrew)
Brevity is the soul of wit (Hamlet)
Refuse to budge an inch (Measure for Measure / Taming of the Shrew) bad weather)
Cold comfort (The Taming of the Shrew / King John)
Come what come may ("come what may") (Macbeth)
Dead as a doornail (2 Henry VI)
Cry havoc and let slip the
dogs of war (Julius Caesar)
Dog will have his day (Hamlet; quoted earlier by Erasmus and Queen Elizabeth)
Devil incarnate (Titus Andronicus / Henry V)
Eaten me out of house and home (2 Henry IV)
Elbow room (King John; first attested 1540 according to Merriam-Webster)
Faint hearted (I Henry VI)
Fancy-free (Midsummer Night's Dream)
Fight till the last gasp (I Henry VI)
Forever and a day (As You Like It)
For goodness' sake (Henry VIII)
Foregone conclusion (Othello)
Full circle (King Lear)
The game is up (Cymbeline)
Give the devil his due (I Henry IV)
Good riddance (Troilus and Cressida)
Jealousy is the green-eyed monster (Othello)
It was Greek to me (Julius Caesar)
Heart of gold (Henry V)
'Tis high time (The Comedy of Errors)
Household words (Henry V)
A horse, a horse! My kingdom for a horse! (Richard III)
In a pickle (The Tempest)
In my heart of hearts (Hamlet)
In my mind's eye (Hamlet)
It is but so-so(As You Like It)
It smells to heaven (Hamlet)
Kill with kindness (Taming of the Shrew)
Laughing stock (The Merry Wives of Windsor)
Laugh yourself into stitches (Twelfth Night)
Live long day (Julius Caesar)
Love is blind (Merchant of Venice)
Melted into thin air (The Tempest)
Though this be madness, yet there is method in it ("There's a method to my madness") (Hamlet)
Milk of human kindness (Macbeth)
Much Ado About Nothing (title)
Murder most foul (Hamlet)
Neither rhyme nor reason (As You Like It)
Not slept one wink (Cymbeline)
Once more into the breach (Henry V)
One fell swoop (Macbeth)
Out of the jaws of death (Twelfth Night)
Own flesh and blood (Hamlet)
Star-crossed lovers (Romeo and Juliet)
Parting is such sweet sorrow (Romeo and Juliet)
[What] a piece of work [is man] (Hamlet)
Play fast and loose (King John)
Pomp and circumstance (Othello)
Pound of flesh (The Merchant of Venice)
Primrose path (Hamlet)
Quality of mercy is not strained (The Merchant of Venice)
Salad days (Antony and Cleopatra)
Seen better days (As You Like It? Timon of Athens?)
Send packing (I Henry IV)
Snail paced (Troilus and Cressida)
Something in the wind (The Comedy of Errors)
Something wicked this way comes (Macbeth)
Spotless reputation (Richard II)
Such stuff as dreams are made on (The Tempest)
The short and the long of it (The Merry Wives of Windsor)
Sweets to the sweet (Hamlet)
There's the rub (Hamlet)
This mortal coil (Hamlet)
Too much of a good thing (As You Like It)
Truth will out (The Merchant of Venice)
Wear my heart upon my sleeve (Othello)
What's done is done (Macbeth)
What the dickens (The Merry Wives of Windsor)
Wild-goose chase (Romeo and Juliet)
The world's my oyster (Merry Wives of Windsor)
Yeoman's service (Hamlet)
http://www.pathguy.com/shakeswo.htm