NavList:

Bob Goethe, you asked:
"Around when was the 32-point compass phased out in favor of compasses with a 360° face?"

Very roughly, in the first decades of the twentieth century. Of course there were compasses with degrees much earlier, and there are compasses with points today. So at best we could consider a date when one style was "predominant" compared to the other. Equally important, like many things in navigation, there is a cultural element. The subject and practice of navigation are not monolithic. Different practices, styles, and histories are found in different countries and in different sectors within those countries. Navigation practice aboard ocean-going ships in the British Royal Navy is certain to be different both in broad practices and specific technical details from those of commercial fisherman in coastal waters of New England, for example.

It's a worthy topic for some basic research, accessible to almost anyone. How could you answer this? Primary sources would be difficult to find. Some logbooks may hint at it. Diaries and journals probably include occasional references. But good luck, right? Finding useful references would not be easy, given how few such documents have been rendered in searchable digital form. Among secondary sources, navigation manuals can help, but you have to be exceedingly careful here since the manuals and handbooks often include chapters copied wholesale from earlier publications. Frequently chapters on basic navigation remain unchanged for decades.

Navigation manuals and also nautical almanacs (taken as a general category) have another data source besides the main text that's often overlooked: advertising. Not everything sells, so advertising can mislead by displaying "potential". And the most mundane and common styles of instruments need no help in the market, so advertising can skew toward the exotic. But at least it gives us a general impression, and they are often visual. Photos and in earlier decades, drawings and etchings, are usually prepared from the most recent products since that's where the money is. And in fact, that's the key: follow the money. Advertising tends to reflect something approaching reality on the ground and on the sea, because new compasses cost money.

Frank Reed