Many years ago when I first rode a racer on roads, I made a right turn on the right most lane because I was too short and needed the road divider when I stop. I find this way of making right turn unnecessarily dangerous as you need to keep left after the turn, and worse if the vehicle on your left want to make a U-turn.
Later, I made right turn on the left most right turning lane so that I will end up on the left lane after the turn. However ... , you need to take the lane for those lanes that allow both forward moving and right turning. Even when the lane only allow right turn, beware of drivers who still wanted to go straight. However, on junctions where there isn't a right turn only signal, it is quite dangerous as you have to judge the traffic and trust that your bike won't break down half-way crossing.
Both ways described above require filtering to the right, which may not be possible on heavy or fast traffic roads. I strongly discourage the first method and do not encourage the second method.
When friends carried me on their motorbikes/scooter in Taiwan, I realized that they seem to be disallowed to travel on the outer lanes (i.e. left lanes in Taiwan). So when they needed to make left turn (corresponding to our right turn), they would have to do a L-shape route. If it was green, they moved to the opposite and wait for green on the perpendicular direction. If it was red, and crossing allowed, they would cross over, and cross again and turn to the left.
Since then, unless at junctions that I am very familiar and find it very safe to use the second method, I would be using the L-shape method. However, when cycling across pedestrian crossing, do it at walking speed and watch out for traffic situation. Especially for the 2nd L-shape method. In Singapore, right turning vehicle can turn when there are no on coming vehicles and pedestrians are still crossing the road. Drivers determine no pedestrians with the assumption that pedestrian travel at walking speed. Thus, if one runs or cycles across at significantly higher speed than walking, collision is very likely.