Different scapular have different requirements for enrollment and are tied to different devotions/causes. I don’t know that any require to be invested on specific days though some do have indulgences attached to the date of investiture (i.e. gold and violet Scapular of St. Joseph).
The two most common are the brown scapular of Our Lady of Mt Carmel and green scapular of the Daughters of St Vincent de Paul. Any priest may bless and invest the brown scapular. The feast day normally associated with it would be July 16th. The FSSP parish in my neck of the woods is dedicated to Our Lady of Mt Carmel and invites anyone not already enrolled in the confraternity to do so on that feast day. The green scapular does not require enrollment, but should be blessed. I don’t know of any traditions for this one, but September 27th is the feast of St Vincent de Paul.
The other scapular I’ve seen in “the wild” is the 5 fold scapular. It is interesting in the fact that 4 of the 5 scapular would normally require a priest of the order to invest someone or permission granted by the order’s supperior. When combined into the 5 fold scapular it can be invested by any priest.
