Frank,
This thread was a "hotbed" of controversy over a year ago.
My proceedure is this:
I ALWAYS inspect the interior FIRST. This way, the only "dirt" on my shoes is that which gets picked up from my car to the front door.
I always "wipe" on whatever mat is present.
I always wear shoes that have a "soft" bottom, so that the chance of my slipping on flooring is minimized, and so that the probability of scratching hard flooring surfaces is minimized.
If I perceive that the removal of shoes is a hard and fast LEGITIMATE cultural custom, I offer to remove my shoes. The owner is typically 50/50 on this---half say "yes" and half allow the shoes to remain. Remember: even for those homes where it is the custom to remove shoes, they Do Not go into the basement area without shoes.
Finally, the reason for not removing my shoes is SAFETY: MINE. We are predominantly self-employed, and one day not working is a day without income. If , while without shoes, I slip and fall or step on a sharp object and wind up with an infection (or worse), I will not be able to continue to make a living. Sue the homeowner? Sure, but how long does that take, and how successful will you be?
My shoes are on 98% of the time.