Edible lollipop sticks for a complicated birthday cake
The brainstorming in the comments has provided numerous examples:


Candy cigarettes

These are likely the most realistic sticks you will find, as they're usually completely white, but fairly soft so use care when attaching the muffin to the top:


Pocky / Mikado

These are chocolate covered biscuit sticks, they're much more sturdy but not the right colour:



There is also a Cookies &amp; Cream flavoured Pocky, marketed as Pocky Panda, which is white and the closest resemblance they make. This may require being purchased from the internet though as I'd never heard of them before:


Rolled Chocolate Wafer Biscuits

These are simply tubes of wafer biscuits, typically filled with chocolate, such as DeBeukelaer Pirouline and Pepperidge Farm Pirouette:


Stick Candy and Rock

Quite simply sticks made out of candy, I suspect Rock will be less common outside the United Kingdom, but Stick Candy should be simple to find.

Rock is a hard candy tube up to an inch thick which could support a table, nevermind a cupcake:



Stick Candy is basically un-bent candy canes and less thick than rock, though very similar: