As in years past, I will briefly discuss the nominees in the most popular categories and then give a pick and a prediction for each. My pick is the film I feel deserves the award most out of those nominated. My prediction, on the other hand, comes strictly from an analysis of which nominee is most likely to win.
Before I start, though, there are a couple of matter to be addressed. First off is the awarding of the Question Mark Award (for the most dubious nomination) to Chris Noonan, the director of "Babe". Although this little gem came out of nowhere to grab oodles of critical acclaim and audience regard, Noonan just isn't (yet) Best Director material. The second matter of business is the usual whining for films and film artists which should have been nominated but weren't. "12 Monkeys" should have been nominated in both the picture and screenplay categories while Annette Bening was overlooked for her standout turn in "The American President". Now on with the nominees:
Picture - "Il Postino (The Postman)" has the distinction of
having the smallest audience of any of the Best Picture
nominees, in addition to being a foreign-language film.  For
this reason it should be considered the longest shot in the
bunch.  "Babe", which seems to be an unlikely choice, is a real
wild card now that it has earned the Golden Globe for Best
Comedy or Musical.  "Braveheart" and "Apollo 13" are just the
kinds of films that the Academy loves to recognize being,
respectively, an epic period saga and a true-life account of
American bravery.  Rounding out the nominees is an adaptation of
the Jane Austen novel "Sense and Sensibility".  This one is a
British costume drama, the kind of film that gets nominated
nearly every year but never takes home the Oscar.  It's about
time they did.
Actress - Sharon Stone is the only nominee in this category who
can be written off.  Although she proved to the world with
"Casino" that she can act, her past films will weigh heavily in
the Academy's decision.  Susan Sarandon, for "Dead Man Walking",
is currently 0 for 4, but is a constant favorite with critics
and audiences alike.  Elisabeth Shue gave a powerful performance
as the hooker with the heart of gold in "Leaving Las Vegas", but
she has never been a critical darling before this point in her
career.  Meryl Streep, for "The Bridges of Madison County", is a
solid choice and one who has been recognized by the Academy in
the past.  Finally, there is Emma Thompson who, as good as she
was in "Sense and Sensibility", is more likely to garner the
screenplay award for that film.
Actor - Richard Dreyfuss, for "Mr. Holland's Opus", is nobody's
favorite in this category filled with stronger nominees. 
Massimo Troisi, for "Il Postino (The Postman)", may be a
sentimental favorite because he died just hours after completing
the film.  Sean Penn in "Dead Man Walking" and Anthony Hopkins
in the title role of "Nixon" both deserve to be recognize, but
Nicolas Cage is Tinsel Town's "golden boy" of the moment and
will likely take top honors.
Supporting Actress - Joan Allen and Kathleen Quinlan for,
respectively, "Nixon" and "Apollo 13", both had small, thankless
roles in their films.  Mare Winningham didn't cause much of a
stir in "Georgia".  This narrows down the category to Mira
Sorvino in Woody Allen's "Mighty Aphrodite" and Kate Winslet in
"Sense and Sensibility".
Supporting Actor - Tim Roth was sublime as a sadistic nobleman
in "Rob Roy".  Kevin Spacey hit just the right note as a timid
thug in "The Usual Suspects".  James Cromwell, for "Babe", and
Ed Harris, for "Apollo 13", were good but aren't a match for
their competition.  Brad Pitt, although a bit of a longshot for
"12 Monkeys" is that film's best shot at glory, and surpassed
everyone's estimation of his acting talents.
Director - Chris Noonan won't take it for "Babe" and neither
will Michael Radford for "Il Postino (The Postman)".  Mike
Figgis, for the downbeat and dreary "Leaving Las Vegas", stands
a chance, as does Tim Robbins for "Dead Man Walking".  But the
Academy, which is primarily composed of actors, loves to give
the directing award to a fellow actor, such Clint Eastwood for
"Unforgiven" and Kevin Costner for "Dancing With Wolves".  With
the glorious "Braveheart" under his belt, this looks to be Mel
Gibson's year.
Original Screenplay - Woody Allen probably won't win for the
tolerable "Mighty Aphrodite", and "Toy Story" is an even less
likely choice to take home this award.  "Nixon", although
worthy, is about an unpopular person and isn't likely to strike
gold for that reason.  "Braveheart", by Randall Wallace, and
"The Usual Suspects", by Christopher McQuarrie, will likely
battle it out for the gold.
Adapted Screenplay - "Apollo 13", "Babe", "Leaving Las Vegas",
"Il Postino (The Postman)", and "Sense and Sensibility" are the
nominees in this category.  "Leaving Las Vegas" is riding a
groundswell, but it's a film easier to appreciate than to like. 
Emma Thompson should score big here.
Song - Nominated are "Colors of the Wind" from "Pocahontas",  
"Dead Man Walking" from "Dead Man Walking", "Have You Ever
Really Loved a Woman" from "Don Juan DeMarco", "Moonlight" from
"Sabrina" and "You've Got a Friend" from "Toy Story".  Bet
against Disney only if your accountant claims it can be used as
a tax write-off.  And if you're thinking "Toy Story", think
again.
The nominees in the remaining categories are:
Art Direction
    APOLLO 13
    BABE
    A LITTLE PRINCESS
    RESTORATION
    RICHARD III
Cinematography
    BATMAN FOREVER
    BRAVEHEART
    A LITTLE PRINCESS
    SENSE AND SENSIBILITY
    SHANGHAI TRIAD
Costume Design
    BRAVEHEART
    RESTORATION
    RICHARD III
    SENSE AND SENSIBILITY
    12 MONKEYS
Documentary Feature
    ANNE FRANK REMEMBERED
    THE BATTLE OVER CITIZEN KANE
    FIDDLEFEST
    HANK AARON: CHASING THE DREAM
    TROUBLESOME CREEK: A MIDWESTERN
Documentary Short
    JIM DINE: A SELF-PORTRAIT ON THE WALLS
    THE LIVING SEA
    NEVER GIVE UP: THE 20TH CENTURY ODYSSEY OF HERBERT ZIPPER
    ONE SURVIVOR REMEMBERS
    THE SHADOW OF HATE
Film Editing
    APOLLO 13
    BABE
    BRAVEHEART
    CRIMSON TIDE
    SEVEN
Foreign Language Film
    ALL THINGS FAIR (Sweden)
    ANTONIA'S LINE (The Netherlands)
    DUST OF LIFE (Algeria)
    O QUATRILHO (Brazil)
    THE STAR MAKER (Italy)
  
Makeup
    BRAVEHEART
    MY FAMILY, MI FAMILIA
    ROOMMATES
Musical or Comedy Score
    THE AMERICAN PRESIDENT
    POCAHONTAS
    SABRINA
    TOY STORY
    UNSTRUNG HEROES
Dramatic Score
    APOLLO 13
    BRAVEHEART
    NIXON
    IL POSTINO (THE POSTMAN)
    SENSE AND SENSIBILITY
Short Film (Animated)
    THE CHICKEN FROM OUTERSPACE
    A CLOSE SHAVE
    THE END
    GAGARIN
    RUNAWAY BRAIN
Short Film (Live Action)
    BROOMS
    DUKE OF GROOVE
    LIEBERMAN IN LOVE
    LITTLE SURPRISES
    TUESDAY MORNING RIDE
Sound
    APOLLO 13
    BATMAN FOREVER
    BRAVEHEART
    CRIMSON TIDE
    WATERWORLD
Sound Effects Editing
    BATMAN FOREVER
    BRAVEHEART
    CRIMSON TIDE
Visual Effects
    APOLLO 13
    BABE