There are several housing lots around, but all the ones I visit seem to be occupied, with the residents shown as online or offline. There doesn't seem to be any indication of empty lots. How can I get my own?
You should go to the Realty page.
On the left, you can tell which regions have vacant lots. Click one of them, look at the offerings and click to see a house you're interested in.
From that same page, you can buy it, and teleport to it (the first teleport is free.)
Don't forget that having Your Papers In Order is a prerequisite for owning a house.
The easiest way to see what you still have to do to get your papers is this site:
The Glitch Strategy Wiki is also a good resource on housing, housing types, and the way to buy real estate:
Finally, once you've got Your Papers and you know what type of house you want, you'll have to pick an available listing from the Realty page:
Although the devs add housing on a regular basis, houses are in high demand and there might not be a listing that matches what you want. You might have to wait and check for new houses on the market frequently if you're looking for something in particular.
I'll also say that houses are only really useful once you've gotten some levels in caring for trees, crops, and animals. They also have cabinets for storage, but buying bags for your inventory is a better short-term option at lower levels than buying a house in most cases.