Bamboo, one of the hardest woods on Earth, is also one of the most flexible. It is a grass and it thrives mainly in hot tropical regions and in some cold places as well. If you live in a hot, humid region like Panama you don't need to oil shinai due to country's high level of humidity (the rainy season lasts from May to November and rainfall is twice as heavy on the Pacific coast as it is on the lowlands of the Caribbean coast). If you reside in the template European regions you will most likely need to oil shinai. Why? Because in cold weather bamboo goes dry. Too dry. If bamboo is too dry it will quickly loose flexibility reaching point of breakage.
How many times? It depends on your country of residence and local weather. In hot and humid regions almost never, in cold regions once a month or twice.
Type of oil? Linseed seems to be the most popular. But kendoka experiment with all sorts of oils: mineral oil, cooking oil, WD-40. Use what you think works for you in accordance with your type of shinai and your budget. I cannot provide an informed opinion on waxing as I have never waxed shinai. However, Lopez Sensei does recommend waxing to prevent frequent splintering. Wax between bamboo junctures is specially helpful, allowing them to rub smoothly against each other. If you are out of commercial wax you can use candle wax.
Shinai don't break easily as many kendoka believe. You can test this yourself: put on a forearm protector before putting on kote and ask sensei or one of the strongest kendoka to do their most powerful kote. Chances are the protector will bend, crack, even break in little chunks but shinai will remain unsplintered.
It's hard to let go of a faithful shinai. Lopez Sensei had one for nearly 30 years and after all those years of devoted service the wood could no longer tolerate standard practice. Sadly, shinai was disarmed by Sensei and now parts are used for crafting other artifacts such as a makeshift arm protector. You can catch a glimpse of the protector in our YouTube channel, I promise I will post a picture for you.