Professional Traffic Operations Engineer (PTOE) here!
The name "turn signal" implies that you are signaling your intentions to other drivers on the roadway. There is only one way to enter a roundabout in the USA . Since everyone must circulate in a counterclockwise manner, I don't think it is necessary to use a turn signal to enter the roundabout.
Once in the roundabout, the signal would be most helpful if turned on when approaching the leg where you intend to exit. The purpose behind using said turn signal would be to signal the driver at the next entry point that you intend to exit, which would allow them to make a more informed decision about when to enter the roundabout.
If you are in a multilane roundabout, you should
not be changing lanes!! Last year I attended a roundabout class taught by one of the engineers who wrote most of the guidelines on roundabouts (Rodegerts). During the class, we were told that multilane roundabouts should be designed so that you make lane changes prior to entry. Once you enter the roundabout, you should not need to change lanes in order to exit at your desired location (assuming that you entered in the correct lane).
I believe I saw a comment about using a left turn signal to indicate that you are staying in the roundabout, but I think that will just confuse drivers. Signals are used to indicate that you intend to change lanes - either by turning onto an entirely different roadway, or by shifting into an adjacent travel lane. We do not signal other drivers that we intend to stay in a particular travel lane when on a straight section of roadway, or when negotiating a curve. It is expected that you will stay in your travel lane - until you signal otherwise.
Personally, if I saw a driver with their left turn signal on in a roundabout, I would assume they don't know what they are doing - because there are no left turn exits. And, if it is a multilane roundabout, they should not be changing lanes.
------------------------------
Kelly Farabee P.E., M.ASCE
Guyton GA
------------------------------