When using javafx.concurrent.Task, I often want to perform an action when the Task is completed, regardless of whether it succeeded or failed.

I suggest to add new EventType WORKER_STATE_COMPLETED as the common supertype for WORKER_STATE_SUCCEEDED, WORKER_STATE_FAILED, WORKER_STATE_CANCELLED.
This would allow to write

    task.addEventHandler(WORKER_STATE_COMPLETED, e -> handle(e));

instead of

    task.addEventHandler(ANY, e -> {
        if(e.getEventType() == WORKER_STATE_SUCCEEDED
                || e.getEventType() == WORKER_STATE_FAILED
                || e.getEventType() == WORKER_STATE_CANCELLED) {
            handle(e);
        }
    });


Furthermore, I suggest to add method Worker.State.isCompleted() that reutrns true for SUCCEEDED, FAILED, CANCELLED, and returns false otherwise.
This would allow to write

    if(state.isCompleted()) {
        doSomething();
    }

instead of

    switch(state) {
        case SUCCEEDED: // fall through
        case FAILED: // fall through
        case CANCELLED:
            doSomething();
            break;
        default: // do nothing
    }