I am currently working with mathmatical texts from different times and contexts (coding theory and algebra mostly) and therefor the authors use multiple notations for a field.
There is use of $k$, $\mathbb{F}$ (usually as $\mathbb{F}_q$ with Judy Walker), $K$(also defined to have $q$ elements with Reed and Solomon).
I know about $\bar{k}$, but there seems to be no difference in what fields are mentioned.
Is there a good reference of what symbols are "current" to use?
The notation of $k$ for a field comes from German, the German word for field is "Körper". German was the main language for maths in the 1930s but was replaced by English around WW II but people kept and keep calling fields $k$ in some areas of maths. So the notation $\mathbb{F}$ is definitely the more modern one.
For the question of which one you should use, I would stick with whatever is more common in your area. So look at what your advisor does and do the same. This will be easiest to understand for people in your area of math.