A dyson sphere would be a luck discovery, as a properly built one would absorb 100% of it's star radiation and would not be visible from outside at all. Heck even gravitation perturbations would be minimal and essentially undetectable.
Actually there is one way to potentially discover at least the possibility of a Dyson's sphere - and that is if a relatively young and healthy star disappears from our night sky. Either the star-fairy stole it, or somebody built a Dyson's sphere around it

Of course many of the stars we see could already have Dyson's spheres built around them - but we are seeing old light, hundreds of years old in some cases. Light that left the star before the giant world was built around it.
Building such a sphere is perfectly possible - just an immense engineering achievement that would probably take millenia to build. The real difficulty I believe in a civilisation being able to build one is to not destroy itself during the very long time it would take to construct.
To put the task into perspective - imagine the engineering required to build a replica earth shell 20ft thick. That is, a spherical structure the exact size of the earth - but only 20 ft thick and hollow inside.
Pretty massive engineering task, huh? The surface area on the inside of an Earth-To-Sun radius sphere is approximately 1 billion that of the surface area of the Earth. So imagine the length of time involved in getting the materials and building another Earth-sized shell, and multiply that effort to 1 billion to begin to understand what it might take to build a Dyson's sphere.
The reason I use 20ft as an example is that Larry Niven in his essay "Bigger That Worlds" calculated one could build an Earth-radius shell around our sun using only the materials from Jupiter, assuming mass transmutation technologies were in place.
And such a shell would be about 20 ft thick. Of course, it doesn't matter how thick the shell is - the gravitational attraction at any point of the inside of a sphere is zero. And so you'll either need to spin it for centrafugal force and "live" on the band around the equator - using the rest of the sphere to capture energy. Or you would need to work out how to generate gravity artificially.
At 20ft it would be impossible to defend against meteors and comets and so punctures would regularly occur.
Larry Niven's Ringworld presents some advantages over a Dyson's sphere. You can spin it for centrefugal force and the entire structure becomes habitable, building walls on the edge to hold in the atmosphere. A challenge would be to get the energy to spin such a large structure up to the required 770 miles per second to generate 1g worth of centrefugal force outward. But using only the mass of Jupiter you could make such a structure 1000 metres thick and therefore a lot sturdier than a Dyson's Sphere.
But the argument mainly goes that although right now we can't begin to dream of building such megastructures, by the time our civilisation progresses to the stage where we need it, we'll know how to build it.
Right now I am waiting for TransLink in Vancouver to extend their Skytrain from Surrey for the extra 15km or so out to Langley where I live. It was supposed to be done in 2010, then pushed to 2030 - now I hear it may be 2050.
In that perspective it's fair to say that a Ringworld or a Dyson Sphere is a way off yet
