Who first cared about singular points? If you look at the cross $C\subset \mathbb A^2_k$ given by $xy=0$ in the affine plane over the field $k$, you see or compute that it is exceptional  at $O=(0,0)$ for many (obviously not independent)  reasons:   
$\bullet$ The gradient of $xy$ vanishes at $ O$ .
$\bullet$ Two irreducible components pass through $O$.
$\bullet$ If $k=\mathbb C$, the complement of $O$ is disconnected.
$\bullet$ The tangent cone of $C$ at $O$ is not a line .
$\bullet$ The maximal ideal $(x,y)\subset  \mathcal O_{C,O}$cannot be generated by just one   element.
$\bullet$ The sheaf $\Omega_{C/k}$ is not locally free.
$\bullet$ The $k$-morphism $Spec (k[t]/\langle t^2\rangle) \to C$ given by $x=t,=y=t$ cannot be lifted to the overscheme $Spec (k[t]/\langle t^2\rangle) \subset Spec (k[t]/\langle t^3\rangle) $.  
This exceptional character of $O$ is covered by  several negative adjectives: non smooth,non-regular, non manifold-like , singular,... 
Although I know that the purely algebraic condition for singularity (in terms of number generators of the maximal ideal of a local ring) is due to Zariski and that smoothness in terms of infinitesimal liftings is due to Grothendieck, I don't know the earlier history of the concept of singularity.   
So my question  is:
Who first considered explicitly the concept of singularity for varieties , why    the interest and what was the definition? 
Edit
First of all, thanks for the interesting comments. It is  certainly plausible  that Newton knew what a singularity was, but from what  I read (very little) his preoccupation seems to have been classification of curves by degree.
I  am curious about when he or others first wrote down the dichotomy between singular and non singular varieties , in analogy with  Descartes's sharp  distinction between mechanical (=transcendental) curves and geometric (=algebraic) curves ( see here) .
[By the way, if you know French you will be delighted  by Descartes's  old-fashioned but easily understandable language]
 A: Gauss' Disquisitiones generales circa superficies curvas, read at the Royal Society in Goettingen on 8 Oct. 1827, contain many termini like punctis singularibus or singulis punctis. Already on the first page we read cuius singula puncta repraesentare.
Dirichlet's collected works, edited by Kronecker and Fuchs, contain often the phrase "singular cases", and on pag. 365 punctum singulare, (written around 1850).
Riemann's collected works, edited by Weber and Dedekind, contain the first mentioning of "singulärer Punkt" on page 389 that I know of in German. It is taken from a paper about linear differential equations of 1857.
The Earliest Known Uses of Some of the Words of Mathematics supply the following dates:
SINGULAR POINT appears in a paper by George Green published in 1828. The paper also contains the synonymous phrase "singular value" [James A. Landau]. 
Singular point appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young. 
In An Elementary Treatise on Curves, Functions and Forces (1846), Benjamin Peirce writes, "Those points of a curve, which present any peculiarity as to curvature or discontinuity, are called singular points." 
 Additional remark 
Hermann, a correspondent of Leibniz, wrote about singulis locis and singulis punctis in letters to Leibniz of 11 Jan. 1711 and Jun 1712, respectively. [C. I. Gerhardt (ed.): "Leibnizens mathematische Schriften",  Halle (1859) p. 364 and 368]
And we should not forget that l'Hospital's famous theorem (1696) has been designed for singular points of functions only.
