While bike commuting in an urban setting, I've started noticing that some intersections have flashing yellow turn signals during part of the light cycle. What does this mean? If I am on my bike in the turn lane and a flashing arrow appears, can I proceed with the turn or should I wait for the arrow to turn green?
Flashing yellow arrows have been introduced in the US recently to help ease congestion at intersections. This signal should be treated similar to a yield sign or a turn lane not controlled by its own traffic signal. If there is no oncoming traffic, you may make a left turn if it is safe to do so, but you must yield to any oncoming traffic and to any pedestrians in the crosswalk you would cross when turning.
Although laws vary regionally with regards to how bicycles are treated in traffic, in most cases a cyclist would handle this situation the same as any other vehicle. If there is no oncoming, you can make a left turn, but be wary of traffic coming the other way. It may be safer to wait if you don't have a full view of oncoming lanes (i.e. if there is a blind corner or low visibility), since it will take you longer to clear the intersection than a motorized vehicle, and drivers coming the other way will not be expecting a bicycle in the intersection during their green light cycle.
This animation from the Alaska DOT shows how the flashing yellow arrow sequence works:
As always, it might differ from state to state, but at least in CA the meaning of flashing yellow has been established for a long while already: flashing yellow is equivalent to "yield" sign, just like flashing red is equivalent to "stop" sign. The fact that it is an arrow specifically makes no difference whatsoever - it simply means that the signal applies to the specific direction the arrow is pointing to.
When you see a flashing red arrow, it means that you are allowed to proceed in that direction after making a full stop (and, of course, yielding the right of way to the interfering traffic). When you see a flashing yellow arrow, you are allowed to proceed in that direction with no mandatory full stop requirement (you still have to yield the right of way to the interfering traffic).
So, on a busy street the won't be much practical difference between a flashing red arrow and flashing yellow arrow. On an empty street the difference is quite noticeable: yellow arrow allows you to ride right through, while the red one still requires you to make a full stop before turning.
Note that flashing yellow left arrow dictates the same turning procedure that shall be followed in case when no arrow is present at all and the light for going straight is green: you can take the turn after yielding to the oncoming traffic. Flashing yellow just makes it more explicit.
For example, imagine an intersection equipped with red and green arrows for the left turn. People who drive through that intersection often would get used to waiting for the green arrow to make the left turn (with red arrow lit up at all other times). Let's say one day the traffic authorities decided that under some circumstances it is OK to allow a non-protected left turn at that intersection (i.e. a turn without a green arrow). How can they do that? Formally, all they need to do is to turn off the red arrow from time to time (without lighting up the green one, of course). However, many people will still continue to wait for the green arrow purely out of habit. And that's where the blinking yellow helps: it tells them explicitly that they are allowed to proceed after yielding.
For left turning drivers, the flashing yellow arrow means the same thing as a circular green light: Yield to oncoming traffic and pedestrians.
The main difference is the meaning given to drivers who are NOT turning left. The circular green releases other movements in addition to the left turn. The flashing yellow arrow does not. This allows the flashing yellow arrow to be used when the circular indications are red.