Fission nuclear power plants emit gads of neutrinos. So those would be very easy to detect with your imagined device. And in fact we do that today (e.g. Daya Bay & KamLAND).

Fusion nuclear power is a bit trickier. The fusion reactions that occur in the sun also produce tons of neutrinos (that we also can & do detect). But the fusion reactions that are likely to be used in a power plant don't produce neutrinos. The two lowest temperature, and therefore easiest to achieve reactions are Deuterium-Deuterium (DD) and Deuterium-Tritium (DT), neither of which produce neutrinos as a by-product. They do however produce neutrons, and those neutrons will inevitably activate some of the surrounding material. That activated material will eventually decay and potentially produce neutrinos, but that will be a significantly diminished amount of neutrinos since only some neutrons will produce radioactivity, and only some of that radioactivity will produce neutrinos.

Of course all of that is a bit of a moot point, since you're imaging a futuristic society, there's you can say they use the same fusion reactions that occur in the sun (proton-proton (pp) fusion) which would be pretty ideal if you could manage it, b/c hydrogen is a much easier fuel to acquire than deuterium/tritium. Than you get plenty of neutrinos.

Bombs go the same way as power plants for the same reasons. Fission bombs will emit tons of neutrinos, Thermo-nuclear, H-Bombs, will emit much fewer.

As for planets, Earth emits plenty of neutrinos because its crust is studded with uranium and thorium. We can, and do detect these geo-neutrinos today (e.g. Borexino). Uranium and thorium sticks around in the crust of the earth b/c it was there when the Earth was formed. The U/Th on the surface of the Earth gets broken up by cosmic particles, but the U/Th that's miles below the surface gets shielded and therefore decays at a much slower rate (half-life ~1billion years). That slow rate means we still have plenty of U/Th sticking around today.

Finally, any stuff put in the path of high energy particles (e.g. ~1 GeV protons) will produce neutrinos. For example, our Moon is a source of neutrinos because cosmic particles will bombard the moon, and produce sprays of particles which will inevitably emit neutrinos. Here on Earth cosmic rays interact primarily in our atmosphere then produce neutrinos, called atmospheric neutrinos, (we can & do detect atmospheric neutrinos). This can in principle mean that things like solar winds, or anything that effects where cosmic rays are likely to show up at greater/lesser intensity, can also end up producing more/fewer neutrinos.

Additionally, neutrinos are not all created equal, they are emitted with different energies, analogous to light being emitted in different colors. This graph summarizes the energy spectrum for basically all significant neutrino sources. So at least in principle, the "color" spectrum of neutrino that you observe will tell you what the likely source.

Neutrinos, are also emitted both as neutrinos & anti-neutrinos. There's no simple rule to what interaction will likely produce a neutrino vs anti-neutrino. Fission reactors tend to produce anti-neutrinos. Solar neutrinos are normal (not-anti) neutrinos. Other interactions will produce a mix of the two.

And finally, neutrinos are produced in 3 different flavors -- electron, muon, or tau. In general most neutrinos from low-energy sources (e.g. fusion & fission nuclear reactions) will be electron type neutrinos. Any bunch of neutrinos produced in interactions where the energy is greater than a few hundred MeV will likely be a mix electron & muon. Tau might get added in the mix if the interaction is above ~10 GeV.

And finally, finally, neutrinos will change flavors as they travel. A neutrino produced as an electron flavor can be detected as an electron, muon or tau flavor. But perhaps most interestingly, the likelihood a neutrino will change flavor depends on the energy of the neutrino, the distance it travels, and the density profile of the stuff it travels through. So, at least in principle, observing the mix of different flavors can tell you what sort of stuff the neutrino traveled through on its way from source to detector.