What attempts have been done to prove the continuum hypothesis? Of course CH is independent of ZFC but at least Cantor himself tried to prove it. So which attempts has been done in this direction? Are they any trials that are not completely wrong? 
Edit:
I'm interested in results/techniques that have arisen from attempting to prove the CH as Theo said in his comment.I am not interested in wrong proofs of freshmen but in wrong proofs of, for example, Cantor himself.
 A: Your last question is meaningless, given the fact that you know that $\sf CH$ can be neither proved nor disproved from $\sf ZFC$.
Cantor's original approach was ultimately through what is now known as the perfect set property. This essentially means that an uncountable subset of $\Bbb R$ has a copy of the Cantor set sitting inside of it. And of course, if an uncountable set of reals has the perfect set property, then it has to have the same cardinality as the continuum.
Therefore, the line of thinking goes, if every set of reals is has the perfect set property, then the continuum hypothesis must be true.
Cantor showed how to prove that every closed set of real numbers has a perfect subset. And the proof later pushed through the Borel sets, and can even stretch to continuous images of Borel sets, also known as analytic sets. But that's it. No more than that can be proved without additional hypotheses which are themselves not provable from $\sf ZFC$.
This is demonstrated by the existence of Bernstein sets, sets which are uncountable, and while they intersect every perfect set, they contain none. These are sets which definitely do not have the perfect set property. Using the axiom of choice, one can prove that Bernstein sets always exist, and thus came to a close that chapter.
Interestingly enough, it was later proved that it is consistent to have the perfect set property hold for every set of reals, and the axiom of choice of course fails. This was done by Solovay and later improved upon by Truss (the improvement was weakening the choice hypotheses and thus omitting the large cardinal hypotheses). This shows that the line of thinking about the perfect set property is consistent, and therefore not entirely flawed.
