How to interpret a model for the angle of attack, when the translational velocities are tiny? I'm playing with a model for the angle of attack of a flying machine, "M".
If M's angle of attack is modeled by, say, 
$$ \text{AoA} = \arctan \big( \frac{\dot{y}(t)}{\dot{x}(t)} \big)$$
and the flying machine happens to be in a hovering phase, then its translational velocities $\dot{x}(t)$ and $\dot{y}(t)$ will be tiny.  Moreover, some velocities in $x$ are zero for certain times.  
Then, when I plot the AoA during a hovering phase vs. time, there are really sharp spikes upward and downward, while it behaves "normally" in between these spikes.
Qualitatively, how should I interpret these spikes?  It seems that the AoA is spiking upwards and downwards not due to aerodynamic fluid forces, but rather because the velocities are so small that the model for the AoA is reaching its model limits, so that it's a computational issue, not a physical one.  
And indeed, large angle of attacks seem nonsensical: when the flying machine is hovering, it's barely moving, let alone experiencing large angle of attacks and / or torques.
In Matlab, the code doesn't break, because the atan2() function handles a zero denominator just fine.  So in this setting, the AoA can range from $- \pi$ to $\pi$.
Should I consider the spikes, e.g. when the velocity in $x$ is at or near 0, "singularities"?
 A: I would recommend to use atan2. The exact definition can be found here: Wikipedia, Polar Coordinates, Converting between Polar and Cartesian Coordinates. 
$$\operatorname{atan2}(y,x) =
\begin{cases}
\arctan(\frac y x) &\text{if } x > 0, \\
\arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\
\arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\
+\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\
-\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\
\text{undefined} &\text{if } x = 0 \text{ and } y = 0.
\end{cases}$$
The singularity you mentioned is unrelated to you physical model. 
The formula for $\mathrm{AoA}$ using $\arctan$ is just not valid for $\dot x = 0$.
Like you can see on Wikipedia, there are different special cases to consider.
Whenever you have a vector $\mathbf v = (v_x, v_y)^T$ and you want to calculate the angle between $\mathbf v$ and the x-axis $(1,0)^T$, you get this kind of strange singularity, if you are just using one trigonometric function for all possible values of $v_x, v_y$. This is due to the fact, that the triangle with corners $(0,0)$, $(v_x,0)$ and $(v_x,v_y)$ is no triangle anymore if $v_x = 0$.
A: Usually angle of attack is defined by the position of the wing or the craft, not by the velocities.  It just defines how much higher the leading edge of the wing is than the trailing edge.  If you have a helicopter or drone the usual angle of attack tells you how to separate the thrust into horizontal and vertical components.  I don't know the term for the ratio you are working with, but it could be angle of flight or some such. 
You are correct that it gets badly defined when the velocities are small.  It would be good to look at the places this gets used.  Do you use it in any of the control laws, or just plot it?  If you just plot it, you might just refuse to plot it when it is uncertain.  If it is used for control, you may be able to eliminate the error by multiplying your equations by $\dot x$.
