Is it permissible to carry the Mus'haf during obligatory and supererogatory prayers, and where should it be placed during bowing (ruku') and prostration (sujud) if it is permissible to carry it?

Carrying the Mushaf and reading from it during prayer is permissible, but avoiding it is more virtuous to steer clear of scholarly disagreement. The worshiper can place it in their pocket or on a nearby chair when bowing or prostrating. As for bending down to place it on the ground and picking it up in every bowing, this is excessive action that invalidates the prayer.