Storing simple syrup
While a similar question to this one, as @Athanasius points out, I think this question is different enough for a re-visit!

The short story: Though it doesn't require immediate refrigeration, it's best to keep simple syrup tightly sealed and in the refrigerator; it will last longer. Even so, using it within a week to 10 days is best; it might last a month, but best not to keep it for longer. It will eventually grow mold or perhaps other nasties. 

References: Some people say to use within about a week (e.g., Morgenthaler). These people say it will last more like a few weeks or a month (e.g., Food Republic; the kitchn; Kitchen Riffs).

Slightly longer story: a 1:1 sucrose/water simple syrup won't crystallise at refrigerator temperatures (even at nearly freezing, this ratio is nowhere near saturated; reference). That is, there's no disadvantage to storing simple syrup in the refrigerator. In contrast, stronger mixtures like honey may sugar. As an aside, measuring 1:1 by-weight or by-volume are similar result; sugar measures lighter per volume by about 15%.

Stronger syrups (e.g., 2 parts or more of sucrose, 1 part water) will have greater resistance to critters because of the preservative properties of sugar. An anecdotal experiment was done by  alcademics showing increased "freshness" time with increased sugar concentration.

Still, however, this is not canning; you'll eventually get mold or other critters. Since it's not necessary to boil simple syrup to mix (since sucrose is so soluble in water), so (depending on how you make the simple syrup) you're not getting the full effect of sterilization of the storage vessel nor the foodstuff (e.g., as you would with canning jams).