For example: When Trapinch evolves into Vibrava, it's Speed goes from 10 to 70. In a similar way, what Pokemons Base Stat increases the most by evolving?

Magikarp, when it evolves its base stats get raised by 340 points.

As J1309 said

But, there is one more. Feebas, when it evolves its base stats get raised by 340 points.