What can be done about unsightly thread veins found under the surface of the skin?

Yes. Small thread veins, or spider veins / telangiectasia, are easily treated with a variety of treatment depending on their location. On the face, lasers work best. On the legs, sclerotherapy is the most effective. Sometimes these veins on the legs can be associated with larger vein disease and require an ultrasound evaluation.
Yes. Larger veins may require surgical stripping of the veins through incisions in the skin or passage of a laser through the vein and vein valves.
Sclerotherapy. What you are describing sounds like spider veins which can be treated by sclerotherapy or laser. Sclerotherapy works well on the legs and the smaller veins on the face can be treated by laser or even micro needle raidofrequency. You should see a vein specialist for a full evaluation .
Full evaluation. Often thread veins (or spider veins as we call them in the us) are caused by underlying vein trouble that isn't readily apparent to the naked eye. Before any vein treatment, it is important to assess whether there might be abnormal flow (reflux) in veins below the surface of the skin or not. This can be done with ultrasound. Your best bet is to see a vein specialist.
Yes. Most "spider veins" are subcutaneous vessels at come up because of injury or change from pregnancy, hormones, or aging. Generally small veins may be treated by sclerotherapy (injecting a substance that closes the vein) or with laser or high intensity light which coagulates protein which destroys the vein.