I'm actually not going to say a lot on this topic. Since it makes up 66% of the liquid volume of a Manhattan, your choice of bourbon or rye as a basis for this cocktail is largely up to you, your tastes, and the balance on your credit card. Bourbons tend to be on the sweeter, smoother side while ryes tend to be spicier and sharper, given their differing mash bills. But again, it is up to you and what you prefer. The other ingredients are meant to compliment and decorate the dominating flavor of your bourbon or rye.
Unless you specifically ask for a certain variety, bartenders will more often than not select a rye for your Manhattan, and for good reason. The sweetness of the vermouth compliments the spiciness of the rye, and the bitters at the end take some of the edge off the over all sweetness while adding some flavor variety. I'll talk later about vermouths and bitters.
When I ask a bartender to make me their best Manhattan, I'm fascinated by the wide range of whiskey choices they make as a basis for my cocktail. However, I'm therefore less surprised by the wide range of results.