Prepare the meat and stock: In a large stockpot, add the pork shoulder, ribs, trotter (if using), and enough cold water to cover by about 1 inch. Add the onion, garlic, peppercorns, bay leaves, and salt, and bring to a boil over medium-high heat. Once boiling, lower the heat, and partially cover to maintain a very gentle simmer. Cook until the meat is very tender and the stock is flavorful, 2½–3 hours.
Remove the pork from the stock, and set aside. Once cool enough to handle, remove and reserve the meat. Discard the bones.
Set a fine strainer over a large bowl, and strain the stock, discarding any solids. Let cool, then refrigerate for at least 6 hours or overnight, until the fat congeals at the top. Skim away the fat, and reserve for another use or discard.
In a large dry skillet over high heat, add the chiles and cook, turning once, until fragrant and toasted, 20 seconds per side.
In a medium heatproof bowl, add the toasted chiles and enough boiling water to submerge. Set aside until rehydrated and softened, about 20 minutes.
Make the broth: In a high-powered blender, add the chiles (reserve the liquid), onion, and garlic. Blend on high speed until smooth, adding a tablespoon or two of the chile soaking liquid as needed. Set a fine strainer over a medium bowl and pour the chile mixture into it, pressing with a rubber spatula or the back of a spoon to extract as much liquid as possible. Discard any solids.
In a large pot, add the oil and heat over medium-high heat. Once hot, stir in the chile mixture. Lower the heat to medium-low, and continue cooking until thickened slightly, 10–12 minutes. Stir in the reserved pork, hominy, and 6 cups of the prepared stock (reserve the rest for another use). The consistency of the pozole should be that of a thick stew; if needed, add some of the reserved hominy cooking water until you reach the desired consistency. (If using canned hominy, use water or more pork stock.) Continue cooking over medium heat, stirring occasionally, until hot, about 15 minutes.
To serve, ladle the pozole into wide soup bowls. Let each diner garnish their bowl generously with their desired mix of toppings.
To garnish (choose a mix): Corn tortillas, tostadas, or chips, Cotija cheese, Diced firm-ripe avocado, Dried Mexican oregano, Finely chopped white onion, Fresh cilantro, Lime wedges or halves, Sour cream, Thinly sliced green cabbage or iceberg lettuce, Thinly sliced red radishes