Make the vinaigrette: Using a sharp paring knife, peel 2 oranges, following the curve of the fruit; cut between membranes to remove whole segments. Place in a small bowl; set aside. Juice remaining orange into a separate small bowl or large glass measuring cup; whisk in lemon juice, vinegar, and salt. Season with pepper; set aside.
In a small skillet, cook bacon over medium heat until crisp and browned on both sides, about 4 minutes per side. Transfer to drain on paper towels; let cool, and finely crumble.
In a large serving bowl, combine fennel, mushrooms, onion, and spinach. Add reserved orange segments and vinaigrette, and toss to combine. Divide among serving plates, and sprinkle each with crumbled bacon.