Interesting thoughts. For some reason, my mind never thought of raising the shutter speed higher than the inverse of the focal length, ie more than 1/60th second for a 50mm lens etc.
As someone who shoots mostly in available light, I've collected all sorts of tips and tricks to handhold better over the years, although these days am probably out of practice. Mainly I would focus on pulling my elbows tight against my ribs, steady my breathing, gently pressing the shutter on an exhaled breath. If possible, and going really low, which is an 1/8th of a second or less to me, I will try and locate any possible source of support, be that to lean against something, or if I really strike gold, find something to rest the camera on and trip the shutter with the self timer, just as if it were on a tripod.
If the shot is really critical, I'm handholding, and the camera will allow it, I will try to increase my odds of a sharper shot, by putting the camera in burst mode, where by taking three shots in succession I hope the middle one may turn out best.
Any sort of camera support is the best though, and whether resting the camera on something, or mounting it on a tripod, I will always use the self timer mode to trip the shutter. Also on one of my nikon dslrs, there's a mode whereby the camera will raise the mirror automatically a split second before making the exposure, the theory being that by doing so, you are reducing/ eliminating any vibration from the mirror during the exposure. I'm not sure if this has much of an effect, but when on a tripod, I will select it, as it can't hurt.
I hate tripods though, and unless what I am really shooting calls for it, I will either handhold, or try get away with a less cumbersome form of support. Something I bought a long time ago, was a bean bag that has a tripod screw for it to connect to your camera, and have gotten some use out of over the years.
Lastly, the single biggest difference to shooting in terms of sharpness and handholding, has been image stabilisation. As an available light shooter, who tends to shoot static subjects more frequently, image stabilisation has been a game changer. My main dslr lens has it, and I simply cease to worry about shutter speed when using it, until I get down to 1/15th of a second. At 1/15th and 1/8th of a second, I then consider all the above handholding tricks to supplement the image stabilisation.