Singular solution of differential equation

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I have seen the standard examples on this topic. I am a little confused about it. If we are given any random differential equation, how do we identify that it would have singular solution as well? Is it possible that in my method of solving ,I may not obtain the singular solution. I mean the examples I saw had DE in terms of $x,y$ and $\frac{dy}{dx}$ . If I somehow manage to solve by Linear DE form, I won't obtain the singular solutions.
Also can someone give me examples of DE where multiple singular curves are obtained?