Ah, Blast, medical advice you get on the internet ain't worth what you pay for it.

All those posts saying, "My uncle's neighbor's cousin's Aunt Martha swears by using dog spit" aren't likely to be found in the Merck Manual.
First, this message does not in any way constitute medical advice or advocate any course of treatment. (And consult your personal physician if your pieces start turning green or falling off.)
If you search pubmed.gov for "neosporin efficacy," you'll find a mixed bag of results among a total of six papers. If you search for "topical antibiotic efficacy," you'll find 869 papers to add to your summer reading. (Adding the word "wounds" to that reduces the number to a far more manageable 85.)
Here's one quote from Diehr, S., Hamp, A., Jamieson, B., and Mendoza, M. Do topical antibiotics improve wound healing? J Fam Pract. 2007 Feb;56(2):140-4.
A clinical trial compared the efficacy of a cetrimide, bacitracin zinc, and polymyxin B sulfate gel (a combination not available in the US) with placebo and povidone-iodine cream in preventing infections in 177 minor wounds (cuts, grazes, scrapes, and scratches) among children. The antibiotic gel was found to be superior to placebo and equivalent to povidone-iodine, in that it reduced clinical infections from 12.5% to 1.6% (absolute risk reduction [ARR]=0.109; 95% confidence interval [CI], 0.011–0.207; NNT=11).2
A double-blind study of 59 patients found Neosporin superior to placebo ointment in the prevention of streptococcal pyoderma for children with minor wounds. Infection occurred in 47% of placebo-treated children compared with 15% treated with the triple-antibiotic ointment (NNT=32; P=.01)They also note honey works well (and I think it tastes better).
I'll add these studies seem quite inadequate given the very small sample sizes. Clearly, larger studies are called for, and I wonder if too many people seal wounds (e.g., with goop and bandages) before properly cleansing them. Saline solution is your friend. I keep some stream saline nose spray in my pack. (Not the drops or mists.)
Have fun and let's be careful out there.