Operational Calculus and Distortionless Transmission
There's an example of Heaviside style operational calculus in the link I posted to above. One of the reasons I became interested in this subject again is that I was getting into electronics and I wanted to simplify computations of properties of simple linear circuits. I had this crazy idea that capacitors and inductors could be treated as resistors whose resistance is differential operator valued. Turns out that this wasn't an original move. This is exactly what Heaviside did well over 100 years ago and it was the secret weapon he used for much of his work. He could solve a wide array of ordinary and partial differential equations with ease. Very briefly, his idea was to write the differential operator d/dx as the symbol p and then treat p much like a conventional algebraic variable. He turned differential equations into ordinary algebraic equations.
A great example of this was when he studied the electrical signal that emerges from a long cable as a function of what was sent into the other end. If W is the outgoing signal, and V is the incoming signal, he showed that in his model, W = √(A+Bp)/√(C+Dp) V, for some constants A, B, C and D, that depend on the properties of the cable. At first sight this is meaningless - what is the meaning of the square root of a differential operator? Heaviside had ways to deal with these things, but that's not what he did here. He noticed that if he picked A, B, C and D such that A/B=C/D then he could cancel the p from top and bottom. The net effect was that if this condition held, the signal emerging was the same as the signal entering (apart from a time delay). In physical terms this meant that adding inductance to a long cable would allow it to carry the signal without distorting it. His contemporaries had been declaring long-distance telegraphy impossible because inductance would distort the signal, but here was Heaviside suggesting that inductors be added. The British Post Office ignored Heaviside's claims and it was left to a physicist in the US to put his ideas into practice - ideas that today formed the backbone for the nascent global telecommunications industry. Heaviside couldn't even get much of his work published because mathematicians like Burnside (boo! hiss!) rejected it as unrigorous. Needless to say, Heaviside died a bitter neglected old man...
Foreshadowings of Special Relativity
I'm fascinated by some of the theoretical clues about relativity that were appearing before Einstein. There were obvious results like the Michelson-Morley experiment and the Lorentz-Fitzgerald contraction proposed to explain it. But there were clues in other places too. HG Wells, in The Time Machine, said "There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it", so we already have a popularisation of the idea of a symmetry between space and time. Heaviside spent much of his time working with Maxwell's equations (which should really be called Heaviside's equations) which inherently has Lorentz group symmetry. This means that any physical predictions made from Maxwell's equations must also have Lorentz invariance. As nobody had explicitly recognised this as a symmetry of nature at the time, it meant for some unusual seeming results. For example, at the time Heaviside was working, the notion that the electromagnetic field stored energy was becoming popular. Heaviside compared the field of a static charge and a moving charge and noticed that for the same charge, the latter stored more energy. This meant that to accelerate a charge required putting extra energy into it which would go into the field. In other words, a charge should feel like it has more mass than it has. The apparent mass contained a familiar 1/√(1-v²/c²) factor and so he noticed that this mass increase grew as the charge's velocty approached that of light. In particular he noticed that the mass would become infinite at the speed of light, exactly as predicted by Special Relativity. Heaviside was never deterred by anything as trivial as an infinity so he went on to study the properties of superluminal particles and predicted and derived the properties of what should be called Heaviside radiation.
(BTW When Heaviside tried to study the geometry of the field around a moving spherical charge he initially made a few mistakes that were eventually fixed by someone else using Heaviside's own methods correctly. One thing that was noted was that the spherical symmetry was flattened. Yet another hint of Lorentz-Fitzgerald contraction.)
I'd love to also say something about Heaviside's battles with Preece and Tait because they are highly entertaining. But instead, I just recommend reading the book for yourself.