How to find Singular solutions of differential equation based physical model? To get singular solutions, do we always need a guess or experiment?
Can we get it from a relation of family of curves of general solution? For example, $(y')^2-xy'+y=0$ has the general solution $y=cx-c^2$. It has a singular solution of $y=x^2/4$, too. If you draw family of curves of general solution (a bunch of straight lines) as well as curve of singular solution (a parabola), you can find parabola is touching general family of curves with a pattern. Can that be a point to get singular solution?
In general, is there a way to calculate singular solutions mathematically?
 A: 
To get singular solutions, do we always need a guess or experiment?

No. You consider a differential equation where the derivative is not given explicitly, i.e., your equation is
$$
F(x,y,p)=0,\quad p=\frac{dy}{dx}.
$$
The condition $F(x,y,p)=0$ gives you a surface $\pi$ in the space $(x,y,p)$. It is not difficult to figure out that your singular solution (which is actually called a discriminant curve) is given by the projection of the set of points of $\pi$, where the tangent plane is vertical.
A very readable account of the underlying theory can be found in Arnold's book, Section 1.3. 
A: *

*Whenever you take any function of y(x) in the denominator, make sure that zeroes of that function are included in general solution found by solving equation. If not then they are part of singular solution. 

*If your singular solution is envelop to general solution, you can find it by maximizing( or minimizing) general solution y(x) keeping x constant and derivating wrt arbitrary constant. For e.g. general solution y=cx−c2, maximizing it wrt c gives c = x/2. Putting c gives singular solution y=x2/4.

