Symbol for Euclidean norm (Euclidean distance) Which symbol is more commonly used to denote the Euclidean norm: $ \left \| \textbf a \right \|  $ or $ \left | \textbf b \right |$?
As mentioned above, I don't know what is most common (statistically). However, ff you have a vector $V$ space over say the real numbers $\mathbb{R}$, then you can have a norm $\|\cdot\|$ on the vector field (so you get a normed space). One thing that you would like is:
$$
\| \alpha v\| = \lvert\alpha\lvert\cdot\| v\|.
$$
for $\alpha \in \mathbb{R}$, and $v\in V$. Here the single vertical lines is the norm on the real numbers and the double lines is the norm on the vector space.

If you consider for example the real numbers as a vector space over itself, then you can use the absolute value as a norm.

If you have the vector space $V = \mathbb{R}^n$ as a vector space over the real numbers, then I do believe that the standard notation is the doube lines $\|\cdot \|$. Again, this is because you want to have the single lines for the real numbers. Note that even though the absolute value and the norm seem like the same thing, they are different because the absolute value is evaluated at real numbers, the norm of the vectors. Indeed the Euclidean norm is defined from the absolute value. So for $v = (v_1, \dots, v_n)$, you have
$$
\| v \|^2 = \lvert v_1\lvert^2 + \dots + \lvert v_n\lvert^2.
$$ 
Note what happens when $n = 1$, then you have $\|\cdot \| = \lvert\cdot\lvert$.
$ \| \mathbf{v} - \mathbf{u} \| $

or

$ \| \mathbf{v} - \mathbf{u} \|_2 $