Can little blue veins on my legs go away or am I stuck with them?

Sclerotherapy. The good news is you are not stuck with those blue veins. Depending on their size, they can be very effectively treated with sclerotherapy (for small veins), phlebectomy (for med size veins) and endovenous ablation for the larger veins. All of these procedures can be done in an outpatient office. See an experienced vein specialist for evaluation so they can determine which treatment is best!
Varicose Veins. Small varicose veins can be treated with injections or lasers. Such treatment are often not covered by insurance as they are considered cosmetic. They tend not to go away by themselves. They may recur. Large varicose veins often require surgical treatment by a vascular specialist. Sometimes such treatments are covered by insurance. Wearing support stockings can reduce progression.
Sclerotherapy best. Sclerotherapy is less expensive, less painful and more effective than lasers for small leg veins. Large bulging veins should be evaluated with duplex ultrasound, with treatment based on results of the exam. Stripping has been replaced by much less invasive treatments such as endogenous thermal ablation or foam sclerotherapy. Large bulging surface veins are often treated with ambulatory phlebectomy.
They can be treated. You can have vein treatment which is called sclerotherapy. This causes the veins to be closed, sclerosed by 2 -3 sessions and the blood is diverted to healthy veins. Sometimes if you have symptoms of pain, swelling, cramping, restless legs, or ankle veins associated with these surface veins, you may need an ultrasound of your legs first to treat an underlying medical process first: phlebology.Org.