A side from the same cuisine scores best. Italian, French and Mediterranean mix; Mexican and Latin go with American; Asian mains get Asian sides or plain cooked vegetables.
One oven, four burners
Dishes share the oven only within 25°F of each other, a roasting pan fills the oven, and the stove has four burners. A side that cooks on the stove while the main roasts scores best.
Ready together
A side is ready inside the main's cook time, or it can be made ahead. The dessert is made ahead, or takes an hour or less.
Balance
No second starch next to pasta, rice or potatoes; a main without one gets one. A rich main never gets a rich side, and fresh, sharp sides score higher next to it. A side never repeats the main's star ingredient.
Every recipe page for a main lists its sides and dessert under Goes well with, with the reason for each.