>>1813393works for commuter towns, but not scalable to actually dense cities, because bringing a bike on means you can only fit 1/3 of the usual number of passengers (assuming everyone is bringing a bike).
here in taipei, you can take bikes on the regular bento trains in the luggage compartment, if you have one, or you can take non-folded bikes on the MRT (but only off-peak, only on the lines that aren't light metro, and you're banned from the most crowded interchanges)
