Roughly speaking how stable are short sections of single-stranded RNA in exposed environments compared to double-stranded DNA?
The main difference in stability is due to the very high amounts of RNase in pretty much all fresh biological materials. Naked RNA is easily degraded by just touching it with skin. The first step in most RNA extractions is often to add an RNase inhibitor.

For example, from Wikipedia's Ribonuclease; RNase contamination during RNA  extraction:


  The extraction of RNA in molecular biology experiments is greatly complicated by the presence of ubiquitous and hardy ribonucleases that degrade RNA samples. Certain RNases can be extremely hardy and inactivating them is difficult compared to neutralizing DNases. In addition to the cellular RNases that are released, there are several RNases that are present in the environment. RNases have evolved to have many extracellular functions in various organisms.