If we're going to get really serious (again) about seeking out signs of intelligent, technological life in the universe, it's critical that we develop a comprehensive overview of options. I'd not for a moment suggest that this is an original thing to propose, but I do think that we all tend to get excited by flavor-of-the-moment ideas for technosignatures, rather than evaluating the greater landscape of possibilities. The catch, of course, is that this landscape can be pretty extraordinary, stretching our credulity to the point where one is inclined to shrug and say 'whatever'.
The underlying challenge is to ask whether we can really rank possibilities with only our human priors to guide us - which could be entirely inappropriate. I'm not sure we know the answer to this, which is all the more reason to have more people thinking about the problem.
So, for some fun I thought I'd throw together my own list of ideas for intentional and unintentional technosignatures. None of these might be manifest anywhere in our galaxy (or even universe), but some could, and perhaps all of them are. More importantly, this list doesn't get too deeply ensnared in any caveats or assumptions about the minds and motivations behind such things, or about their agency.
Let's start with conventional possibilities.
Low-frequency electromagnetic radiation:
What we'd commonly call 'radio'. Long wavelengths are probably useless for cosmic distances, with wavefronts severely distorted by the interstellar medium, but anything from frequencies of a few megahertz up to Terahertz (sub-mm wavelength) bands seem fair game. SETI has long favored frequencies like 1.4 to 1.7 GHz as a cosmic 'watering hole', encompassing hydroxyl and hydrogen natural emission frequencies and a comparatively quiet spectral range for Earth-based listening. It's a logical first choice, but in my opinion we actually have no idea whether this is really the optimal place to listen.
Electromagnetic radiation can also carry information in polarization of its E and B fields. In this case a continuous carrier wave might look completely uninteresting, but in fact be wildly gesticulating in its polarization properties over time.
Mid-frequency electromagnetic radiation:
Infrared, through visible light, to ultraviolet radiation all rain down on us constantly from the rest of the universe and could be manipulated for signaling. Optical SETI is all about this range, especially coherent, lased emissions. Perhaps deliberate signaling, perhaps leakage from interstellar light sail launches, perhaps incomprehensible experimentation or cultural expression.
High-frequency electromagnetic radiation:
Soft X-rays to hard X-rays to gamma-rays also wash across us from the cosmos. We've gotten very good at detecting rapid variations in bright sources like pulsars and even active galactic nuclei. Experiments like NICER have demonstrated how X-ray data on pulsars can even be used for a type of interplanetary and interstellar GPS system.
But with all electromagnetic signaling/communications/leakages there are innumerable options for how this could actually work. We tend to start with the notion of point-to-point information exchange, or arbitrarily broadcast information that arrives as photons direct from the source. I think this is merely one set of options.
Laser-painting & Bat-Signals:
For example, got a (really) big laser? Then 'painting' cosmic structures might be a way to study them, and a way to broadcast widely. Interplanetary LIDAR could wash single frequency light across entire planetary systems, lighting them up in scattered photons. Similarly, why not illuminate a nearby cloud of interstellar gas and dust like a giant Bat-Signal? A rapidly modulated patch of light from a nebula would attract attention from multiple directions for the price of one laser.
Laser beacons and transit related signaling:
If you know someone can see a planet transiting its parent star you can set up to signal at precisely the right moment and in the right direction. You can also (as my colleagues David Kipping and Alex Teachey have pointed out) mask or alter what some alien astronomer will infer about your planet.
Big mirrors (really big mirrors) with active controls could easily send signals: on/off, partially on/partially off and so on. Arrays of many smaller mirrors would offer the most versatility. Inadvertent signaling could come from efforts at terraforming (or more generally enviro-forming) or power generation utilizing a local star. But there could also be more subtle effects from any distribution of reflective structures across planetary surfaces or in interplanetary space - the 'glints' of technology.
The best form of detectable matter that is largely unperturbed in passing through the universe? It's probably neutrinos. Pulsed neutrino beams would be an excellent way to propagate information with relatively little ambiguity about their artificial origin.
Anomalous isotopic mixtures in stellar photospheres or the interstellar medium:
Deliberate or inadvertent sprinkling of artificial isotopes or elemental mixes into stellar atmospheres or interstellar regions would be a very different kind of technosignature. Perhaps there's an extraterrestrial Long Now organization out there somewhere imprinting information onto its parent star for posterity?
Stellar activity manipulation, and stellar population manipulation:
Coronal and chromospheric activity on stars might be altered (unintentionally or otherwise). Magnetic field interactions could perhaps be a mechanism for doing this. Manipulation of stellar behavior could conceivably produce anomalous stellar populations - either in elemental composition or deviation from expected main sequence behavior or late-stage evolution.
Cosmic rays arrive at Earth all the time, and some have traveled vast distances. Even unstable species of subatomic particles, from neutrons to muons and more, can be preserved through relativistic time dilation. But ordinary old atomic nuclei also do just fine. Zipping around at close to the speed of light they could encode data by their atomic number or arrival sequence. We just have to figure out how to capture them or retrace their original composition after they collide with Earth's atmosphere to produce showers of other particles.
Molecules designed to produce specific spectral signatures in interstellar medium or planetary system gas:
As astronomers know, the more complicated a molecule is the more difficult it is to identify in interstellar space through the emission or absorption of radiation. Big molecules end up producing lots of complicated band-structures in spectra as they vibrate and rotate in space. But perhaps it's possible to manufacture molecular structures tuned to generate very specific spectra - encoding information or simply shouting out 'artificial'.
Transit or occultation of compact stellar objects - white dwarfs and neutron stars:
The strangely varying brightness of star KIC 8462852 (also known as Tabby's star after the astronomer Tabetha Boyajian) prompted speculation on partial eclipse of the star by some kind of complex structure that might be artificial (an alien mega-structure). So far that doesn't seem likely for this one case. And if you wanted to send deliberate messages this way you'd be better off eclipsing a smaller object where you get more bang for your material expenditure - it's easier to occult or partially occult a white dwarf star (1 or 2 percent the size of our Sun) or a neutron star (a few tens of kilometers in size) with a lot less material.
Matter dumping onto compact objects and black hole accretion systems:
Why build massive power generating systems for interstellar communication when you could dump small amounts of matter onto objects like neutron stars, or into accretion structures around black holes? A few kilograms of matter simply dropped onto a typical neutron star will release the equivalent energy of a several megaton yield hydrogen bomb. Correctly timed release of mass onto one of these systems would allow data transmission.
The largest detonated hydrogen bomb, the Tsar Bomba, has been estimated to pump out a luminosity equivalent to about 2.1x1017 Joules per second (for a fraction of a second). Our Sun's luminosity holds steady at about 3.8x1026 Joules per second. That's a billion times more luminous. But that's actually not too bad. These days we're homing in on directly imaging exoplanets that are roughly a billion times fainter than their parent stars in narrow band luminosity. So letting off a whole structured sequence of very big thermonuclear devices across your planetary system might be a way to say hello - albeit briefly.
This is not a new idea at all, but the transport of something like DNA wrapped up in a protective membrane would be a way to send information, or simply reveal the presence of life with very alien properties. Such microscopic messengers could perhaps be blown between stars by laser-propulsion. Or placed billions of years ahead of planetary system formation for later incorporation. It could be a molecular message, it could also be a piece of bio-code that would insinuate itself into other life and make its presence known in some other way. And as wildly speculative as it is (and unsatisfying in terms of scientific origin stories) there is a chance that seeding life on another world is the ultimate act of extraterrestrial messaging.
Artifacts and/or storage devices (the Fed-Ex option):
It's still true for most of us that if we want to move a serious amount of data around it's more efficient to load a hard drive and have it flown around the world than to send it on the regular internet. Bandwidth across interstellar distances may be equally constrained. Better to package up information materially and fling it between the stars. It might take 100 years to physically send a petabyte of data across 10 light years, but that's an equivalent data rate of 10 terabytes a year or about 317 kilobytes a second. Not bad throughput for a distance of 60 trillion miles.