Many times I'm in Shul and don't have a place to sit for Tachanun. What's proper procedure in such cases?
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Lean your head on your arm while standing. At the end of the first paragraph (usually the transition from seated-leaning to seated-normal), stop leaning; at the start of the next paragraph (usually the transition from seated-normal to standing), just stay standing.
Sitting is the preferred position for head-leaning, but not required for it.
If one leans against something to the point that if it were removed he would fall it is considered as if he were sitting. Will give you source when I get a moment.