The first thing I noticed was how common a circular shape was, over and above any other shape. There were squashed circles too (ellipses). Squares were quite often present (encasing a circle). Some badges were versions of the traditional shield shape which was also common. I suppose because shields are culturally very recognisable as a badge of allegiance. Triangles are seen, not as frequently as I first thought.
Next most common was the logos made up of words. A fairly common trope is a quite complex image plus the name of the company. The image is usually a very easily recognisable shape such as an animal.
I explored a logo for a yoga company. There are 2 types of imagery that are commonly associated with yoga, firstly a body shape of a person doing a typical pose and secondly a lotus flower which has very distinctive shape. I found one logo that had turned the shape of a meditating person into a single line.
The body shapes offer a lot of potential for a good logo as they are very distinctive and can be simplified even further and still look like a body shape. I turned the poses I had found into single lines. Some were more successful than others, some start to look ambiguous and would need refining in order to work. Some poses stop looking like yoga poses but some are very characteristic and so would work well as a logo.