I recently came across something new to try. Humming helps exchange air between the nose and the sinuses.
Here we show that nasal NO [nitric oxide] levels increases dramatically during humming compared with normal quiet nasal exhalation. This effect is likely due to increased contribution of NO from the paranasal sinuses. Humming causes the air to oscillate, which in turn seems to increase the exchange of air between the sinuses and the nasal cavity.http://www.atsjournals.org/doi/full/10.1164/rccm.200202-138BC#.WAbKSlQrIdX
This exchange doesn't occur if the ostia are completely blocked, though. (https://www.ncbi.nlm.nih.gov/pubmed/15305890)
In 2006, a researcher in Sweden wrote a review on how humming affects ventilation between the sinuses and the nose. (https://openarchive.ki.se/xmlui/bitstream/handle/10616/38896/thesis.pdf?sequence=1) He mentions a case study where a man was able to clear out his sinuses by humming only: http://www.george-eby-research.com/html/chronic-rhinosinusitis.pdf. The case study was done by a man who used to sell zinc as a cure for colds, so I'm not sure how helpful it is, but it's interesting.
Humming has been tested and found to increase improvement of allergic rhinitis (hay fever) symptoms in patients who were given an intranasal steroid spray. (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3392338/) Unfortunately for my curiosity, the institute ethical committee didn't allow the researchers in that study to have a third group that only did humming without any steroid treatment, so it's unclear whether the benefit from humming was from increased delivery of the steroid spray to the sinuses or from the mere act of humming.
Next time I feel my sinuses start to become clogged, I'll try humming. Humming seems harmless, and I might as well do what I can to keep the sinuses open before they become blocked and cause sinus headaches.