During any part of daavening you shouldn’t hold things that you don’t want to drop. This includes even during Pesukay D’zimra or Yakum Purkan. (ShA 1 – MB 1,2,5)
It is likewise forbidden to seat a young child next to you during tefilla. (MB 4)
Whenever it is a mitzva to hold something, you may hold it during daavening. e.g. siddur, lulav on Sukkos, Sefer Torah during Birkas Hachodesh. (ShA 2 -MB 2)
If a sefer falls on the floor and you cannot take your mind off it, you are allowed to go pick it up when you finish the beracha that you are saying. Otherwise, do not interrupt. (MB 7)
Do not communicate with your fingers and the like during tefilla. (MB 7)
If you get mixed up during Shemoneh Esray you are allowed to walk over to get a siddur. (MB 7)
It is proper for everyone to have a siddur open to listen well during Chazoras Hashatz. (MB 9)