Lens designs are created with a balance of trade-offs in cost, weight, audience, volume, etc. Some apertures tend to occur more often than others because they represent points of balanced trade-offs, and often because these are the best apertures a manufacturer can offer due to physics and size constraints.
At the most basic level, lens focal length divided by maximum aperture determines how big the front element needs to be, which then influences cost, weight, and potentially speed and/or ease of construction. A 50mm f/1.0 lens needs to have a front element at least 50mm wide to admit enough light, whereas a 50mm f/2.0 lens only needs a front element at least 25mm wide. This in turn means the inner elements will be proportionally smaller, and smaller lens elements drastically improve final weight and construction complexity. As a heuristic, the smaller the lens elements, the cheaper the lens will be to produce.
In addition to constraints by physics, sometimes the synergy of an entire lens line is considered. One such aspect is the filter ring size in front of a lens. One example of a well-considered line are the classic Nikon AI/AI-s lenses, where many of the lenses had a front filter thread of 52mm.
Here's a sample of focal lengths and maximum aperture, along with corresponding front-element minimum size. Note that all of them fall under 52mm, and you can find variants that have a 52mm filter thread.
Lens Front Element Min
Nikon 35mm f/1.4 AI 25.0mm
Nikon 50mm f/1.2 AI 41.7mm
Nikon 85mm f/1.8 AI 47.2mm
Nikon 135mm f/3.5 AI 38.6mm
Nikon 200mm f/4.0 AI 50.0mm
In the autofocus/DSLR era, Nikon has moved on from the limits imposed by adhering to a 52mm filter thread size, and you will find that they and other camera manufacturers have lens offerings whose filter thread sizes tend to cluster around 67mm, 72mm, etc. Given the filter thread and focal length, you can often estimate with a good margin of error what the lens offering will be in terms of maximum aperture for most lenses between 15mm and 200mm.
At smaller apertures, the main constraint is often autofocus capabilities. Many camera bodies in the DSLR age to date have a hard autofocus sensitivity limit of f/5.6, f/7.1, or f/8, at which they will cease to autofocus, and some bodies have a mix of AF points that span that range of minimum aperture. When considering that AF availability is important in order to sell camera bodies in volume (and also provide ease-of-use), this limits the bottom end of a lens range.
For example, it is rare to find any enthusiast or consumer zoom lens that is slower than f/5.6 wide open.
Because the front-element rule still applies for light gathering even for zooms, the short end will always have an equal or better maximum aperture on a zoom, and given the focal range for the zoom, you can see how many zooms cluster around f/4.5, f/3.5, and f/2.8 at the wide end (shortest focal length).
Finally, f/4 has one special property that makes it a hard limit for some lenses, particularly long/telephoto lenses. When using a lens with a teleconverter, every 1.4x multiplier on focal length costs one the same amount of maximum aperture light gathering capability. Thus, f/4 with a 1.4x teleconverter brings it to f/5.6; f/4 with a 2x teleconverter brings it to f/8. When combined with autofocus limits, it's no surprise that a lot of premium super-telephoto lenses are offered at f/4 or wider. Both Nikon and Canon have 500mm and 600mm f/4 offerings, 300mm and 400mm f/2.8 offerings, and 200mm f/2 offerings (or better, in the case of Canon). When combined with 2x teleconverters you will get 1000mm/1200mm @ f/8, 600mm/800mm @ f/5.6, and 400mm f/4. Similarly, only in very extreme cases will you find anything longer than 800mm (both Nikon and Canon offer a f/5.6 here). The weight and the cost of providing lenses beyond these apertures drastically narrows the potential range of customers that are willing to buy them, as well as those who can make effective use of their capabilities.