[OutsiderFan]

Any team you play every year has to be considered a rival, but there are degrees, like:

1. Are your games meaningful in the context of settling championships?
2. Do you recruit against each other?
3. What's your proximity to each other?
4. How even are the results?
5. Are there incidents that have inflamed either side, that have happened beyond the actual final score results?

Going by this measure, I'd rank them like this, giving 1 pt to each team in terms of Stanford rivalry intensity:

1. USC - 1, 2, 3, 4, 5 - 5 pts: It's a clean sweep for the Condoms.  This is easily Stanford's greatest rival.  Games are very meaningful, they recruit against each other, NorCal-SoCal, they are usually tight games, they trade wins, what's your deal / USC cheating in sports and elsewhere.

2. Notre Dame - 1, 2 (double), 4, 5 - 4pts: The games have been quite meaningful, particularly for ND playoff chances that Stanford has snuffed out 2/3 times in last decade, they recruit hard against each other, the games are usually super close, and Stanford fans really hate Notre Dame claiming parity with Stanford academically, while still adopting a win at all costs attitude Stanford fans loathe, manifested in perception of Brian Kelly.    

Note: It probably shouldn't be a surprise that the top two are private schools. There is something about this that makes the rivalry with these two just a bit more meaningful due to Stanford also being private.

3. Oregon - 1, 4, 5 - 3 pts: Meaningful games, usually tight/nobody dominates,  Stanford fans don't like the criminals Oregon recruits. You might say Oregon is more important than Notre Dame, but I guarantee most Stanford fans would pick beating Notre Dame as more satisfying than beating the Ducks, if they could only have one win between the two.

4. Cal - 2, 3, 5 - 3 pts: Recruit against each other, 30 miles apart, Cal-Stanford academic rivalry. These teams haven't been consistently good enough at the same time, and their games really have never settled anything in the Pac-12.

5. Washington - 1, 2 - 2 pts: They do settle championships, recruit a lot of the same players in PNW, but they haven't been close games or really been any kind of back and forth, aren't close, and Washington has always been pretty clean.

6. UCLA - 2, 3 - 2 pts: Recruit against each other, NorCal-SoCal, but the games have largely settled nothing over the years and Stanford has ownage over last decade.

7. WSU - 1 - 1 pt:  The Cougs have won some games against Stanford lately, and some have been important in terms of settling division, but this is just not firing anyone up as a rival like the others.

8. OSU - 0 - 0 pts: Sure the Beavs are in the North, but I don't ever remember these teams playing for anything meaningful ever. I'd imagine the Beavs occupy a similar place fo most Pac-12 programs.

[OutsiderFan]

Any team you play every year has to be considered a rival, but there are degrees, like:

1. Are your games meaningful in the context of settling championships?
2. Do you recruit against each other?
3. What's your proximity to each other?
4. How even are the results?
5. Are there incidents that have inflamed either side, that have happened beyond the actual final score results?

Going by this measure, I'd rank them like this, giving 1 pt to each team in terms of Stanford rivalry intensity:

1. USC - 1, 2, 3, 4, 5 - 5 pts: It's a clean sweep for the Condoms.  This is easily Stanford's greatest rival.  Games are very meaningful, they recruit against each other, NorCal-SoCal, they are usually tight games, they trade wins, what's your deal / USC cheating in sports and elsewhere.

2. Notre Dame - 1, 2 (double), 4, 5 - 4pts: The games have been quite meaningful, particularly for ND playoff chances that Stanford has snuffed out 2/3 times in last decade, they recruit hard against each other, the games are usually super close, and Stanford fans really hate Notre Dame claiming parity with Stanford academically, while still adopting a win at all costs attitude Stanford fans loathe, manifested in perception of Brian Kelly.    

Note: It probably shouldn't be a surprise that the top two are private schools. There is something about this that makes the rivalry with these two just a bit more meaningful due to Stanford also being private.

3. Oregon - 1, 4, 5 - 3 pts: Meaningful games, usually tight/nobody dominates,  Stanford fans don't like the criminals Oregon recruits. You might say Oregon is more important than Notre Dame, but I guarantee most Stanford fans would pick beating Notre Dame as more satisfying than beating the Ducks, if they could only have one win between the two.

4. Cal - 2, 3, 5 - 3 pts: Recruit against each other, 30 miles apart, Cal-Stanford academic rivalry. These teams haven't been consistently good enough at the same time, and their games really have never settled anything in the Pac-12.

5. Washington - 1, 2 - 2 pts: They do settle championships, recruit a lot of the same players in PNW, but they haven't been close games or really been any kind of back and forth, aren't close, and Washington has always been pretty clean.

6. UCLA - 2, 3 - 2 pts: Recruit against each other, NorCal-SoCal, but the games have largely settled nothing over the years and Stanford has ownage over last decade.

7. WSU - 1 - 1 pt:  The Cougs have won some games against Stanford lately, and some have been important in terms of settling division, but this is just not firing anyone up as a rival like the others.

8. OSU - 0 - 0 pts: Sure the Beavs are in the North, but I don't ever remember these teams playing for anything meaningful ever. I'd imagine the Beavs occupy a similar place fo most Pac-12 programs.