I know (from experience) that in nusach Ari, any morning that tachanun is omitted, psalms 20 ("Yaancha") and 86 ("hate Hashem ozn'cha") are omitted also, and the chest is not banged when saying the blessing "S'lach lanu". What about when tachanun is omitted for non-calendrical reasons (e.g., in a shiva home or when a sandek is present)? Does nusach Ari include psalm 20? psalm 86? the banging of the chest?
The Siddur of the Alter Rebbe writes:
"Minhag Sefarad: Any day on which Tachanun isn't said, Lamnatzeyach and Tefilla LeDovid aren't said either. For example, the whole month of Nissan, Peisach Sheini ... or any day on which their is a Bris in the Shul, or a Chosson for the [seven] days of partying" (emphasis mine).
These aren't calendrical days.
I'm not sure about Chabad, but Sepharadim either include all of the following or omit all of the following. They never say only some of them:
Two other parts of the service have the exact same conditions on when they are said as the above three parts, with the added factor that they are only said on certain days of the week.