Reading Dyson's original argument gives some useful information. He says that such a sphere would have a surface temperature of 200-300 Kelvin. That's a tiny fraction of the surface temperature of a star. Such a sphere would also have to be rather large - he says that it would have to have a circumference the size of Earth's orbit. So we would simply have to look for a massive, cool object radiating very little visible light yet curving space-time just as a star would. Not an easy job, but not impossible.
Besides a black hole, scientists don't know of any other naturally-occurring objects with similar properties. Neutron stars would be insignificantly small, as would other compact stellar remnants (large black holes notwithstanding, but even they could emit different types of radiation). So a Dyson sphere would look completely different when compared to all other objects we know.
Now, Dyson rings, swarms, etc. would be much different. It is possible that a Dyson device consisting of many smaller objects - such as a ring or swarm - could be mistaken for natural objects occurring in a stellar system, such as a dense protoplanetary disk. However, I'm not sure just how much light from the central star such a swarm or ring would let through (quite a lot, I imagine); this could make it easier to distinguish such a device from a typical protoplanetary disk.
Edit
I should add that I don't know of any current or planned experiments searching for Dyson spheres; these are simply methods that could be used.
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