Applications of Modular Symbols

Enumerating all elliptic curves of conductor N.
Computing a basis of modular forms of given weight and level.
Proving theorems towards the BSD conjecture; e.g., that L(E,1)/Omega is a rational number.
Computing the modular degree. Understanding the modular degree well <==> ABC conjecture.
Loic Merel's "Uniform boundedness theorem".

Enumerating all elliptic curves of conductor N.