It is not possible to draw tangent at all the points of a singular curve. What is the specialty of this and how it is related to cryptography and elliptic discriminant?
Not-so-useful answer: An elliptic curve is by definition a non-singular curve. Therefore by definition we use non-singular curves in elliptic-curve cryptography.
Why not use singular curves? It turns out that the group structure on those curves is isomorphic to the multiplicative group of a (quadratic extension of) a field. Therefore the discrete logarithm problem is not harder than of that in a finite field, so there is little reason to use the curve in the first place.
The discriminant characterizes all curves which are non-singular. In other words, the curves that are non-singular are exactly those for which the discriminant $\Delta$ is non-zero.