Grease a shallow 2-quart casserole dish. Place bread in casserole; sprinkle with cherries.
In a medium bowl, mix eggs until well beaten. Add rum, if using, cinnamon, and salt, whisking until blended. Stir in eggnog. Pour eggnog mixture over bread, pressing bread into custard.
Cover and refrigerate bread mixture for 30 minutes or overnight to allow bread to absorb custard, stirring once.
Preheat oven to 350 degrees. Bake bread pudding, uncovered, 50 to 55 minutes or until center is almost set. Cool in pan on wire rack for 15 minutes.
Just before serving, place ice cream in small microwaveable serving bowl; heat in microwave oven on High 30-35 seconds, stirring once. Remove bowl from microwave and stir until smooth, but still thick and cool.