Before moving to Singapore I had never really thought about taking my shoes off when entering a house or a place of work. Shaking your host firmly by the hand and looking them in the eyes yes, but slipping off my shoes before entering their home or place of work was the last thing on my mind. In truth, it seemed more in keeping with the behavior of a burglar than an invited guest.
In every country you visit in Asia, the most common cultural aspects they all share are that rice will be a staple part of the diet and that you are expected to take off your shoes when entering a place of residence.
When I first moved here I found this custom quite odd; it would lead to an army of shoes being deposited outside peoples homes and would also require an added sock and pedicure awareness to your wardrobe and grooming schedule. Sadly, my feet strongly resemble those of a Hobbit and are certainly not my favorite body part; alas they are regularly "out on show" here in Singapore.
Aside from the personal embarrassment about the state of my feet, over time I have become far more accepting about this social requirement. I initially presumed that the need was based on a superstition, but soon came to realize that it was simply for hygiene reasons. Shoes are worn outside, outside is dirty, inside is clean ergo you should not wear shoes that you wear outside when inside. It is quite frankly very difficult to argue with this reasoning.
Another quirk to living here is that people often wear face masks in public; this is again an initially peculiar thing to see. Through my Western viewpoint, I initially thought that those wearing such masks were protecting themselves from the perceived disease threat of others. Sadly this assumption says a great deal more about me than them, as the real reason for people wearing these masks is that they are unwell and are protecting others from their germs. Rather than being a selfish act, the wearing of a face mask is actually a very civic minded one.
This is not to say that these masks do not create issues; I was recently introduced to a new starter at work who was obviously unwell and so was sporting a face mask to reduce the spread of germs. A considerate but problematic act on your first day. As I could not see his face on first meeting; I am now unable to identify him within the office. As a result, I have no idea if he has completed his probation period or is even still employed by the firm.
Different customs within different cultures can certainly be peculiar, however more often than not these customs are actually quite sensible and practical. Understanding them can undoubtedly add value to your own customs and social norms. However, this is certainly not always the case, anybody that has attended the alcoholic carnage that is Ascott Ladies Day in the UK or had the misfortune to experience the facilities in a Public Bathroom in France can attest to this. But in these cases, taking off your shoes when visiting someones home and wearing a face mask when unwell, I think it is fair to say that I am very much onside.