What is a "very bad" singularity? In page 94, example 4.42 of Milne's notes on algebraic geometry, he mentions that the singularity at $(0,0)$ is "very bad". 
The surface is $ V: Z^3 = X^2Y $, which has singular locus $ X = Z = 0 $.
What does he mean by this? The tangent space is $ \mathbb{A}^3 $, does this have something to do with the tangent cone?
 A: Without Milne writing something clearer, asking exactly what they meant is going to be a little bit of guesswork - "very bad" is not so precise. 
One interpretation: The point $(0,0,0)$ is in the singular set of the singular set. Often the singular set is nice: we accidentally folded or glued our variety along some subvariety, and if this variety isn't too bad, it may be easy to understand how to untangle our singular variety. Instead, here, in the example of $x^3=y^2z$, the variety we folded over/glued along was singular too - this makes untangling and resolving the singularities of $V(x^3=y^2z)$ a bit more difficult.
A different interpretation would be that the class of singularity that $(0,0,0)\in V(x^3=y^2z)$ is the worst possible. To effectively communicate this, we need some background material. First, we define a resolution of singularities as a proper birational map $\pi:\widetilde{X}\to X$ from a smooth variety $X$. When dealing with a singularity, there are some basic numerical invariants we can attach to each resolution of it, called discrepancies (this wikipedia article gives some basic definitions about what I'm about to explain). We denote these by $a_i$ where $i$ runs over some finite index set. There are five natural boundary regions of these: 


*

*$a_i>0$, called terminal

*$a_i\geq 0$, called canonical

*$a_i > -1$, called log-terminal

*$a_i \geq -1$, called log-canonical

*some $a_i< -1$.


The $a_i$ may change depending on what resolution of singularities we take, but you can never jump from being in one of the above classes to being not in it. There are a good number of theorems about how nice and how much structure $\pi$ must preserve if it belongs to any of the first four classes, but the final class with some $a_i < -1$ is in some sense the wild west: essentially any bad behavior you want can happen if you go looking for it. For example, if you have a resolution with some $a_i < -1$, you can produce a resolution with $a_i < -k$ for any positive integer $k$.
And in fact, the singularities of this surface aren't log-canonical and belong to this fifth class. So this might in fact be what Milne means when writing the words "very bad".
As to your final question about the tangent space and tangent cone: the tangent space will always be "too big" at a singular point - this is part of the definition of singularity. The tangent cone will display some unexpected behavior (in particular, it won't be isomorphic to the tangent space like you would get with a smooth point), and you may be able to calculate some invariants based off the tangent cone that show you some bad or unexpected behavior - so it does have something to do with the tangent cone, but pinning down exactly what is a bit of an adventure.
