When does the 'standard' angular velocity formula not hold? I have read that the formula for angular velocity:
$$\dot {\vec r}=\vec \omega \times\vec r \tag{1}$$
does not hold in some situations, but the book does not specify what situation so please could you produce a list of when this formula does not hold.
If this formula does not hold is it also true that:
$$\vec \omega= \frac{\vec r \times \vec v}{|\vec r|^2} \tag{2}$$
does not hold? 
 A: Electron spin is not the result of a rotation of the electron around itself. In this case, of course (2) also doesn't hold.
In fact, one can show that there is a double implication as follows:
1) if $\vec v$ is defined as in (1) one gets
$$ \frac {\vec r \times \vec v}{r^2} = \vec {\omega} - \vec r \frac {(\vec r \cdot \vec {\omega})}{r^2}. \tag{I}$$
So, as $\vec {\omega}$ is perpendicular to $\vec r$ the equality (2) is implied. 
2) On the other hand if the equality (2) is true it implies
$$r^2 (\vec {\omega} \times \vec r) = (\vec r \times \vec v) \times \vec r = \vec v \ r^2 - \vec r (\vec v \cdot \vec r). \tag{II}$$
So, if your equality (2) is true, and $\vec v$ is defines as tangential velocity, then it implies (1). Therefore if $\vec v$ is defines as tangential velocity, and (1) isn't true, (2) cannot be true, otherwise it would imply that (1) is true.
A: 
I have read that the formula for angular velocity:
  $$\dot {\vec r}=\vec \omega \times\vec r \tag{1}$$
  does not hold in some situations, but the book does not specify what situation so please could you produce a list of when this formula does not hold.

That expression is only true in the case of circular motion. It fails whenever the radial component of velocity is non-zero.


If this formula does not hold is it also true that:
  $$\vec \omega= \frac{\vec r \times \vec v}{|\vec r|^2} \tag{2}$$
  does not hold?

That expression is tautologically true; that's one way the angular velocity of a point mass is defined.
It helps to define a set of unit vectors:
$$\begin{aligned}
\hat r &= \frac {\vec r}{||\vec r||} \\
\hat \omega &= \frac {\vec \omega}{||\vec \omega||} \\
\hat \theta &= \hat \omega \times \hat r
\end{aligned}$$
The above unit vectors are well-defined and are mutually orthogonal so long as $\vec r \times \vec v$ is non-zero. Denoting $r = ||\vec r||$ and $\omega = ||\vec \omega||$, the above yields
$$\vec v = \frac {d\vec r}{dt} = \frac {d}{dt}(r \hat r) = \frac {dr}{dt} \hat r + r \frac {d\hat r}{dt} = \dot r \hat r + r\omega \hat \theta = \frac{\dot r} r \vec r + \vec \omega\times\vec r$$
With this, your equation (1) becomes
$$\vec v = \frac{\dot r} r \vec r + \vec \omega\times\vec r \tag{1'}$$
The above reduces to your equation (1) when $\dot r = 0$.
