Duane brings out the big knives for no-mess bagels:
This one just dawned on me -- I'm surprised I've been doing it wrong for so long. My two girls will often split a bagel for breakfast (among other things). Cream cheese for one, jelly for the other. Neither is really adept at just picking it up and biting it, so I have to cut them into pieces. Usually I use the knife I just used to put on the cream cheese. Bad idea. Cream cheese all over myself as the less-than-firm bagel smooshes under the pressure and I basically end up ripping the thing into pieces with my bare hands.
Forget that. While you're standing at the kitchen counter anyway, just grab your big butcher's knife, the one with the blade that is much larger than the bagel, and do it "pizza style". Four cuts - in half, then half again, then half again twice more, bringing the big blade straight down on the bagel (using two hands to force it with equal pressure). Don't slice. Instead bagel bites. No mess.