My list:
Real Madrid-Barcelona
Man.Utd-Liverpool
Milan-Inter
Are also some other Derbies like Benfica-Porto,Fenerbahce-Galatasaray,Arsenal-Totenham,Boca Juniors-River Plate etc.

Real Madrid vs Barcelona is the best for me

5.Fenerbahce-Gala
4.Schalke-Borussia D.
3.Arsenal-Tottenham
2.Boca Juniors-River Plate
1.Real-Barca

Good idea!

Real-Barca and Inter-Milan

Real - Barca
Inter - Milan
Man Utd - Man C. (atm)

Chelsea vs Arsenal
Chelsea vs Manchester United
Chelsea vs Barcelona

Galatasaray vs Fenerbahce
Roma vs Lazio
Manchester United vs Liverpool
Al Ahly vs Zamalek
Boca Juniors vs River Plate
Barcelona vs Real Madrid
Rangers vs Celtic

Boca Juniors-River Plate
Fenerbahce-Gala

Actually... A derby is a rivalry between two clubs from the same goegraphic region. What most of you are talking about are "classicos" (not the actual term) which are plain historic rivalries bwtween the two mos traditional teams in the league.

So best derbies would be:

Lazio - Roma
United - City
At. Madrid - RM
Inter - Milan
Shalke - Dortmund

And the best "clasicos"

(combination of both) Celtic - Rangers and Fenerbahce - Galatasaray

Barcelona - RM
River - Boca
Liverpool - United

@Tuanis: +1

The best derbies if you count in quality of football and players are:

1. United - City
2. Chelsea - Arsenal
3. RM - Atletico
4. Dortmund - Schalke
5. Benfica - Sporting
6. AC Milan - Inter

@Tuanis If you count on these I would say:

1.Arsenal-Spurs
2.ManU-ManC
3.Benfica-Sporting
4.Lazio-Roma
5.Arsenal-Chelsea

Arsenal v Chelsea
Real vs Atletico
Man U v man city
Ac Milan v inter milan

@Tuanis Actually, I was thinking the same. But then I opened up the wikipedia page (Turkish) about derbies. It says " Derbies were considered as a rivalries from the same city, but nowadays with the improving world and football economy; derbies count as big rivalries between teams who have greater economies and media support. "
But I will create my list with the rivalries from the same region.

Derbies with beautiful Football :

1. ManCity - ManU
2. Chelsea - Arsenal
3. Dortmund-Schalke
4. Fenerbahçe - Beşiktaş :P
   Derbies with passion :
5. Fenerbahçe - Galatasaray
6. Boca Juniors - River Plate
7. Lazio - Roma

Is it this time of the year already? ( old members will get this)

I'd like to make two points clear.

1. Derbies are only between teams of the same city (like @Tuanis said).

2. 'Clasico' only refers to Real Madrid - Barcelona matches. When you call other matches (like Liverpool - United) 'Clasicos', it takes away the importance and meaning of the word. Especially considering we are speaking in English and the word itself is Spanish.

^El clasico may resemble RM - Barcelona match but because it is a one of a kind type of fixture then the term is adapted to the equivalent of that match in many other countries. We dont have a term for a game like Liverpool - United because Derby can mean something else, a different kind of rivalry.
In Latin America it is very common to call the biggest game in the league the Clasico, so it's not only a Real Madrid v Barcelona thing, but Im not quite sure if that match was the one that originated the term.

Juventus - Torino

Best derbies i love watching :
1- Barça - Espagnol
2- RM - Atletico
3- Man United - Man City
4- Chelsea - Arsenal
5- Milan - Inter

Best rivalries i love watching :
1- Barça - Real
2- Bayern - Dortmund
3- Man City - Chelsea
4- PSG - Om

I dont really care wich derby is the best, but those are the ones i prefer to watch lately.

Essendon vs Collingwood ......sorry, wrong type of football.

^
AFL ???

@Eden. Yep AFL. Massive game of the year played on Anzac Day which attracts an average crowd of around 85000