Can specific Weierstrass curves be some benefit from Montgomery/Edward form?
Curves P-256 and Secp256k1 cannot be converted to Montgomery or Edwards forms because they are of prime order. 

Let's take the example of Secp256k1 and try to convert it to a Montgomery curve:

A Montgomery curve $by^2=x^3+ax^2+x$ has a point of order 2. Such a point verifies $2P=\mathcal{O}$ or think of it equivalently $P=-P$ which means $(x,y)=(x,-y)$ so these points are on the $x$-axis. It is obvious then that the point $(0,0)$ on a Montgomery curve has order 2. Now given the secp256k1 curve $y^2\equiv x^3+7 \pmod p$ where $p=2^{256}-2^{32}-2^9-2^8-2^7-2^6-2^4-1$, we are looking for points on the $x$-axis which means those that satisfy $x^3 \equiv -7 \pmod p$. There is no solution to this modular cubic equation.

In general curve P-256 and Secp256k1 cannot be converted to Montgomery form because they have a prime order ($\ne 2$) and thus they cannot have a point of order $2$ (otherwise $2$ would divide a prime number beacause of Lagrange's theorem).

The same reasoning works for Edwards curve $x^2+y^2=1+dx^2y^2$ because the point $(0,-1)$ has order $2$ and the points $(\pm 1, 0)$ have order $4$.