Check out Jane Smiley's A Thousand Acres, which is a fantastic retelling of King Lear. It's set in modern times, with a farmer who divides his land among his three daughters, but it's complicated by a history of family abuse. In King Lear, it's unclear why Lear's daughters Regan and Goneril are so cruel to their father and only the youngest, Cordelia, is kind to him, and the audience is left to assume that there is something inherently wrong with G & R. In A Thousand Acres, however, Smiley adds a history of family abuse. The older sisters in Smiley's book were abused by their father, so they have a real conflict with their father, and a legitimate reason for being unkind. The youngest daughter, on the other hand, was not abused, and she has a much easier relationship with their dad--and like Cordelia, can't understand why her sisters are so unloving.
There are plenty of works based on, or modernizations of, Shakespeare's plays. The more important thing to note is that there are allusions to Shakespeare everywhere. You can look up any Shakespeare play in Wikipedia and you'll see a list of allusions in literature and popular culture.
James Joyce's Ulysses has countless allusions to Shakespeare. (They are listed, by play, in Weldon Thornton's Allusions in Ulysses.)
Hamlet's Dresser, by Bob Smith, is an autobiography that tells of the author's lonely and sad childhood and recalls the way in which Shakespeare's works changed his life.
Gertrude and Claudius is a novel by John Updike that examines the story of Hamlet from a different perspective (focusing on the relationship of the title pair).
O is a 2001 film whose story parallels Shakespeare's Othello. The language and situations are modern, but the story is almost identical.
The first thing I thought of when I saw your question was, of course...Star Trek VI: The Undiscovered Country. This line comes from Hamlet's "To be" soliloquy. in which he is pondering whether to live or to die. He asks, who would bear all the grief and burdens,"To grunt and sweat under a weary life, But that the dread of something after death, The undiscovered country, from whose bourn No traveller returns, puzzles the will, And makes us rather bear those ills we have Than fly to others that we know not of?"
Death is the beginning of eternity, life after death--the undiscovered country--the place no one knows enough about to risk suicide and the damnation that might bring.
Movies quote Shakespeare lines or make references to his work all the time. Music does the same--lots of Romeo and Juliet references.
One other place which is always making literary allusions is Gilmore Girls, though I can't think of a specific Shakespeare reference offhand.
One novel that uses a Shakespearean allusion both titularly and thematically is Aldous Huxley's Brave New World. The title comes from The Tempest, and in the play, it's meant to be taken at face value, praising the new world "that has such people in't." Yet in Brave New World, John the Savage uses the reference ironically. He's been dragged into a world that is full of madness and immorality in his eyes. For him, it becomes almost a hell on earth. There's several othe Shakespearean allusions in the novel as well. John reads to Helmholtz from Romeo and Juliet (at which he laughs, unable to understand the emotions between the young lovers). He also imagines himself Othello, seeing Lenina as a betraying Desdemona.
The West Side Story is a musical written to mirror Romeo and Juliet.
Ten Things I Hate About You is a modern-day satire to Shakespeare's classic Taming of the Shrew.
Many authors have used Shakespeare's words in their titles. William Faulkner used The Sound and the Fury. Agatha Christie used By the Pricking of My Thumbs, and Ray Bradbury used Something Wicked this way Comes.
But perhaps a really good work for you to research is "The Lovesong of J. Alfred Prufrock" (1917) by T.S. Eliot. He alludes to a Prince Hamlet character for his main character. This shows the main character struggling with a difficulty and realizing he cannot struggle through a situation as well as the character Hamlet in Shakespeare's play Hamlet.
How to Read Literature Like a Professor is a book worth checking out that has many of these allusions.
Cole Porter's "Kiss me, Kate" is an old stage musical about a theatre company trying to put on their musical version of The Taming Of The Shrew, while, backstage, similar themes play out in the actors' personal lives.
I've always liked it. You can check out Bob Fosse's film version or, on youtube, you can see a great staged version here: http://www.youtube.com/watch?v=9d6DMbtKxdc