Seems like the last few years I've been getting into the classics like:
Cups and Balls
Torn and Restored Newspaper
Egg Bag
Torn and Restored Card
Linking Rings
Slydini's Silks
Gypsy Thread

Cut and Restored Rope. Rising Card. Card Tricks (4 Aces, Selected Card Revealed). Producing Silks. More. And as the old catalogs said, "Limited only by your imagination."

I would add chop cup to the list. It’s a classic and is part of many repertoire.

Hanging Coins - David Roth would be on my list. Monumental step forward in Coin Magic.

The history of the Card Sword and the Card Stab.
Free eBook (PDF)
http://maigret.org/cardsword/cardsword.html

Stuff magicians like to play with seems to go in and out of fashion apart from what working performers keep in their repertoires.

For coin items - the HPC trick and copper & silver transposition go on the list.
For card items - the cards across, oil and water, and ace assembly go on the list.
getting into parlor/stage - the milk pitcher w/wo the lightbulb in the lamp is a list item
the 20th century silk trick (the silk winds up tied between two others)
...

This is a very interesting question. First, we have to decide what qualifies a trick to be a classic? Looking in the dictionary there are three definitions which seem pertinent:


1. Judged over a period of time to be of the highest quality and outstanding of its kind.
2. Remarkably and instructively typical.
3. A work of art of recognized and established value.

Subjectively putting those things together in magic speak, we are looking at tricks which are:

1. Of the highest quality - fools, surprises, and entertains audiences.
2. Over a period of time - has withstood the test of time, say, 50 years or more
3. Outstanding of its kind - best of breed/quintessential for one or more fundamental effects of magic
4. Remarkably and instructively typical - we as magicians can learn from studying them
5. A work of art of recognized and established value - used by performers of the highest level, A-quality material

In addition, there is the question of venue (parlor/platform, stage, close-up). Based on this, here is my list with one each for the most common fundamental effects (and a grand mention, too):

Parlor/Platform

1. Production: Pulling a Rabbit out of a Hat (or box, etc.)
2. Vanish – Milk in Cone
3. Transposition – Twentieth Century Silks
4. Transformation – Color Changing Silk
5. Multiplication – Multiplying Billiard Balls
6. Penetration – Linking Rings
7. Restoration – Torn and Restored Newspaper
8. Animation – Rising Cards
9. Prediction - Invisible Pack - mental epic is a close contender
10. Escape - Siberian Chain Escape


Stage

1. Production: Flower Act (e.g. Blackstone's Enchanted Garden, Rolf Dinardi)
2. Vanish – Sword Basket
3. Transposition – Metamorphosis (i.e. Substitution Trunk)
4. Transformation – Lady to Tiger
5. Multiplication - Multiplying Bottles
6. Penetration – Cutting a Women in Half
7. Restoration – Cut and Restored Rope, especially Walking Knot
8. Animation – Dancing Handkerchief
9. Prediction - Confabulation/Prediction Chest (really hard to separate "trick" from "effect" for these)
10. Escape - Straight Jacket Escape

Close-up

1. Production: Coin production (e.g. coin from ear)
2. Vanish – Silk vanish in hand
3. Transposition – Ambitious Card (Copper, Silver, Brass is a strong contender too)
4. Transformation – Color Changing Knives
5. Multiplication - Sponge Balls/Bunnies
6. Penetration – Matchbox penetration
7. Restoration – Gypsy/Hindu Thread
8. Animation – Haunted Deck
9. Prediction - Invisible pack
10. Escape - Thumb Cuffs (one of the few escapes done closeup)

Grand Mention: Cups & Balls - This is a rare trick in which routines include the first six fundamental effects almost equally; production, vanish, transposition, transformation, multiplication, penetration. So it really is on a level of its own as being a "classic" of magic.

Edward Claflin & Jeff Sheridan's book Street Magic is a very good read in terms of the earliest effects