Understanding metric tangent cones. Just here trying to grasp better the concept of metric tangent cones in a metric space (in the sense of rescalings). And maybe said better, I'm trying to understand a bit more the 'bad' behaviours of these cones. The main problem being that my intuition outside nice smooth spaces is very poor. 
Specifically, let $(X,d)$ be a metric space. And call $(Y,d_Y,p_y)$ a tangent cone of $X$ at $p$ if it is the pointed Gromov-Housdorff limit of the pointed spaces $(X,\lambda d,p)$ for some sequence $\lambda\to\infty$.
Now, it is clear to me that such a limit might not exist. Its already a bit less clear that a tangent cone might exist but it need not be unique. I think I can come up with some examples of this last behaviour but more examples would definitively help. So,
1) Could someone please explain some simple and maybe not so simple examples of non-unique tangent cones? Maybe a simple examples, but also for instance examples where the cones are not homeomorphic? can this happen?
Next I was wondering how bad things can get. For instance, the tangent cone at a point might be completely different from cones of nearby points. But, 
2)are there examples, for instance, where the cones are different (not iso, not homeo?) for every point inside some neighbourhood. (some similar behaviour?)
and lastly,
3) Any examples of interesting 'bad' behaviours that would be nice to keep in mind?
I would be really greatful if someone could help me built a bit more of intuition on these matters. Thanks! 
 A: A fun, albeit quite tame, example of non-uniqueness of tangents is given by a spiral in the plane that converges very quickly in the radial direction. Consider the spiral $$[0,\infty)\to\mathbb{R}^2,\quad t\mapsto (r(t)cos(t),r(t)sin(t))$$
where $r$ is a decaying to zero faster than exponentially, take for instance $r(t)=e^{-e^t}$. When you zoom in on this curve, you will see every half-line starting from 0 as a possible tangent along some sequence of zooming factors.
But this is just an example where uniqueness of tangents fails only at one point, and all the tangents are even diffeomorphic (as they are all half-lines).
However in general, if you put no restrictions on your metric space $(X,d)$, basically anything can happen. See for instance the paper "Locally rich compact sets" by Chen and Rossi, where they show the existence of a compact set that has all compact metric spaces with diameter $\leq 1$ as tangents at all points. 
In fact, perhaps more surprisingly, they show that this behavior is typical in the sense of the Baire category theorem. That is, the collection of compact metric spaces that do not have all compact metric spaces of diameter $\leq 1$ as tangents at all points form a meager set. So "most" compact metric spaces have horrendous behavior in their tangent cones.
This is the same sort of statement as the fact that "most" functions $\mathbb{R}\to\mathbb{R}$ are not differentiable at any point.
The construction of Chen and Rossi of a space with the above property is essentially based on the following simple idea. Suppose we want to construct a metric space $X\subset\mathbb{R}^n$ that has at the point $0\in X\subset\mathbb{R}^n$ two distinct metric spaces $Y_1\subset\mathbb{R}^n$ and $Y_2\subset\mathbb{R}^n$ as tangents. For simplicity, I will work with the assumption that $Y_1$ and $Y_2$ are both contained in $B(0,1)$. 
Pick two sequences of scales $\alpha_j\to 0$ and $\beta_j\to 0$ such that they are asymptotically insignificant with respect to one-another. For instance we can choose decreasing sequences such that 
$\alpha_j>\beta_j>\alpha_{j+1}$
for all $j$ and 
$$\lim_{j\to\infty}\frac{\beta_j}{\alpha_j} = \lim_{j\to\infty}\frac{\alpha_{j+1}}{\beta_j} = 0.$$
We construct the space $X$ by scaling down the spaces $Y_1$ to all the scales $\alpha_j$ and scaling down the spaces $Y_2$ to all the scales $\beta_j$. That is we define
$$X = \bigcup_{j\in\mathbb{N}}\Big(\alpha_jY_1\setminus B(0,\beta_j)\Big)\cup \bigcup_{j\in\mathbb{N}}\Big(\beta_jY_2\setminus B(0,\alpha_{j+1})\Big)\subset\mathbb{R}^n.$$
Now consider the unit balls of tangents of $X$ at $0$.
First look at this set at the scale $\alpha_j$, i.e. dilate $X$ by $1/\alpha_j$. Then outside of the ball $B(0,\beta_j/\alpha_j)$ we see exactly the set $Y_1$. But by the construction of our sequences, the balls $B(0,\beta_j/\alpha_j)$ will become smaller and smaller, so along the sequence $\alpha_j$ we see that the unit ball in the tangent cone is the space $Y_1$. A similar argument shows that along the sequence $\beta_j$, the unit ball in the tangent cone is the space $Y_2$.
