As a Salem high grad who also attended Andrew Lewis - I was a freshman the last year AL was a high school - I would say that, overall, Northside is Salem's biggest rival. Primarily due to the two schools being little more than five miles apart, as the crow flies - only Glenvar is closer to SHS. History also plays a part, as Lewis and Northside kids were sparring on and off the playing fields since about 1960. Surprisingly, Lewis and Northside didn't start playing football against each other until 1970, and the Wolverines took four of the five meetings. The only one won by the Vikings was the last game AL played as a high school. It was a miserable way to go out, as Northside prevailed 37-6 in what stood for a long time as the coldest game I ever attended. Over the years, Salem and Northside have had heated rivalries in both football and boys basketball, with the hoops rivalry becoming particularly intense from 1993 to 1996. The Spartans and Vikes butted heads four times in the football playoffs from 1992 to 2000. Believe it or not, from the '70s into the '90s, Lewis and Salem both had somewhat bitter rivalries with Northside in cross country, of all sports. However, once Northside moved to a different district and region, and then a different classification altogether, a little of the juice went out of the rivalry. The two schools still love to beat each other, in all sports across the board. And, the communities still border each other.
If we're talking strictly football only, I would have to agree that Pulaski County is Salem's biggest rival. The second-biggest rivalry overall probably exists with the Cougars. The two have played a lot of high-stakes football games against each other over the decades, in both the regular season and the playoffs (six playoff meetings), and often in front of enormous crowds. I doubt if you could find one old-school Cougar fan who, if they're being honest, doesn't hate Salem with the white hot intensity of ten thousand suns. While still (hopefully) harboring some grudging respect for the Spartan program. And that's fine - a little healthy, non-violent hatred is a hallmark of a good rivalry. Aside from football, PC and Salem also had a lot of spirited clashes on the hardwood in the mid- and late-2000's, several in the district and regional tournaments. There have been occasional big games in other sports through the years, as well. But given the importance of football to the two fan bases, and football's standing as de facto king of high school sports in Virginia, this rivalry is always going to burn a little brighter than those with other schools.
I've never really sensed much of a rivalry with Amherst County. Whatever rivalry there is stems from the ten times the two have crossed swords in the playoffs in football, including five regional finals and one state semifinal. The regular season football rivalry has been one-sided in Salem's favor, as the Spartans have won eight of nine, most in convincing fashion. Granted, a lot of history and emotion has been generated by those playoff games. However, with Amherst County and Salem being an hour-and-a-half from each other, and meaningful games in any sport besides football few and far between, I just don't see much of a rivalry with the Lancers. I think emotions generally just heat up whenever the two collide on the gridiron in the postseason, where Salem holds a 6-4 edge in those ten meetings.
The rivalries with most of the others have ebbed and flowed over the years, seemingly following the fortunes of particular sports. In general, Salem and Blacksburg have had good rivalries in both football and boys basketball for forty-some years, but the rivalries cool as one or the other team is up or down. The relationship between the two fan bases seems, in general, to be an amicable one. Christiansburg, I really don't view as much of a rival at all, due not only to the Blue Demons being in a lower classification, but also to their general ineptitude (over the long term of decades) in the two big revenue sports. Patrick Henry of Roanoke is an old rival, and probably generates more feeling among the older Andrew Lewis crowd (even older than me) than most other opponents. However, PH has almost always been in a different classification, which has eliminated potential postseason clashes with Salem. The football series between the two has been entertaining since starting up again in 2013, after seventeen years of inactivity. The biggest rivalry between the Spartans and the Patriots, over the long haul, has probably been in boys basketball. And Glenvar, the school closest in proximity to Salem, has never been an on-field, on-court rival at all, due to the significant size difference in enrollments (Glenvar is Class 2). I have a feeling off-the-field shenanigans are a different matter, entirely.
Which leaves the two southwest Roanoke County schools. I don't know for sure, but for Salem fans, the most rancorous relationship over the long term might be with Cave Spring. Though standout meetings in football have been few and far between since the Tiki and Ronde Barber days of the early nineties - the Barber-led Knights never beat Salem, by the way - there have been memorable moments, and things always seem heated. The Spartans and Knights have really had a strong, often intense basketball rivalry through the years. Again, it's just an impression, but I'd say there is a lot of bad blood between the two schools' fans. Put simply, these two teams and communities just don't like each other. And then - there's Hidden Valley. For reasons that I will not discuss, as that would lead to absolutely no good, the Titans are the team I personally most enjoy seeing Salem beat. That, despite the fact that the football "rivalry" has been one-hundred percent one-sided (knock on wood) in Salem's favor. When Salem was leading HV 77-7 (yes, you read that correctly) at halftime in 2015, I nearly wept with joy. A not-so-tiny part of me almost wished our coach weren't such a nice guy. Not because of their kids, but entirely due to other, off-the-field issues. That's just the way I felt, and I won't apologize for it. At any rate, I'd say the Titans are rivals with Salem for three reasons: they split off from Cave Spring, and share that old animosity toward all things Spartan; the two are in the same district; and geographic proximity, which is probably the biggest reason. Aside from boys and (occasionally) girls basketball, and girls soccer, there hasn't been much sports-related to base a big rivalry on.