The function singular outputs an expression sequence representing the singularities  of expr.

If two arguments are given and the second argument, vars, is a name or a set of them, the expression, expr, is considered to be a function in vars. If singular is called with only one argument, then expr is considered as a function in the variables returned by the command indets(expr, name).

If a numeric range is given as second or third argument, only the singularities found within this range, if any, will be returned.

Each singular point is represented by a set of equations, the left-hand side of the equations being the variables.

The singular function will return non-removable as well as removable singularities.  For instance, $\frac{x-2}{\sin \left(x-2\right)}$ will report a singularity at $x=2$.

The power of singular to find singularities is basically that of solve. For example, some zeros that solve cannot find may result in singularities that singular will not find.

The singular function may return expressions prefixed by _Z or _N, representing the integers and positive integers, respectively.

$\mathrm{singular}\left(xy+\frac{1}{xy}\,x\right)$

$\left\{x=0\right\},\left\{x=\mathrm{\infty }\right\},\left\{x=-\mathrm{\infty }\right\}$

$\mathrm{singular}\left(\frac{\ln \left(x\right)}{x^2-1}\right)$

$\left\{x=\mathrm{-1}\right\},\left\{x=0\right\},\left\{x=1\right\}$

$\mathrm{singular}\left(\frac{x}{x-y}\right)$

$\left\{x=y\,y=y\right\}$

$\mathrm{singular}\left(\tan \left(x\right)\right)$

$\left\{x=\frac{1}{2}\mathrm{\pi }+\mathrm{\_Z1~}\mathrm{\pi }\right\}$

$\mathrm{singular}\left(\tan \left(x\right)\,1..10\right)$

$\left\{x=\frac{\mathrm{\pi }}{2}\right\},\left\{x=\frac{3\mathrm{\pi }}{2}\right\},\left\{x=\frac{5\mathrm{\pi }}{2}\right\}$

$\mathrm{singular}\left(\tan \left(x\right)\,1..\mathrm{\pi }\right)$

Error, invalid input: singular expects its 2nd argument, vars, to be of type {name, set(name), numeric .. numeric}, but received 1 .. Pi

$\mathrm{singular}\left(\tan \left(x\right)\,1..\mathrm{evalf}\left(\mathrm{\pi }\right)\right)$

$\left\{x=\frac{\mathrm{\pi }}{2}\right\}$

$\mathrm{singular}\left(\mathrm{\Psi }\left(\frac{1}{x}\right)\right)$

$\left\{x=0\right\},\left\{x=\mathrm{\infty }\right\},\left\{x=-\mathrm{\infty }\right\},\left\{x=-\frac{1}{\mathrm{\_NN1~}}\right\}$