I'm setting this answer community wiki so other people can attach their own examples of curves constructed with the methods in this thread.
Now, Qiaochu's suggestion for building a curve $q(y)=p(x)$ with collinear singular points on the horizontal axis amounts to the construction of an appropriate Hermite interpolation problem. More explicitly, one wants to find a polynomial (or rational function) whose first few derivatives at preset points vanish.
In Mathematica for instance, the function InterpolatingPolynomial
can be used to generate a Hermite interpolant. (For systems that do not have such a function handy, the Hermite interpolation problem is solved through either an appropriate modification of the Newton divided differences scheme or by solving an associated confluent Vandermonde system.) The rational interpolant case is a bit tougher, and I am still experimenting with algorithms for the rational Hermite problem so I won't be considering them for now (but might include them in a later edit).
Thus, taking the conditions in Qiaochu's answer, here is how one builds a curve with a node at $(-1,0)$, a tacnode at $(0,0)$, and a cusp at $(1,0)$:
Expand[InterpolatingPolynomial[{{-1, {0, 0, 1}}, {0, {0, 0, 0, 0, 1}}, {1, {0, 0, 0, 1}}}, x]]
The result has fractional coefficients, but you can multiply with an appropriate factor so that all the coefficients are integers, resulting in the polynomial
$$3x^{11}+2x^{10}-15x^9+21x^7-6x^6-9x^5+4x^4$$
Here for instance is a plot of $$y^2=3x^{11}+2x^{10}-15x^9+21x^7-6x^6-9x^5+4x^4$$:
and a more complicated curve, $$y^2-y^3=(x^2+xy+y^2)(3x^{11}+2x^{10}-15x^9+21x^7-6x^6-9x^5+4x^4)$$:
I have yet to make the prescription for getting an isolated point to work, since the curves generated only manage to have separate branches passing through the desired point, but no isolated points at all.
As for T..'s suggestion, the Mathematica code I have requires some serious cleanup, so I shall be editing this answer later to include his parametric construction.
Currently I am trying to find a curve with four-fold symmetry that has four of each of the types of double points. If I manage to find it, I shall be naming it after Qiaochu and T.