Picking fruits from a branch which has fallen:
One may pick fruits off from a branch which has fallen from before Shabbos [and has become completely detached from the tree]. If the branch fell on Shabbos it remains forbidden to pick fruit from it.
Picking leaves from a fallen branch: One may not pick leaves off a branch in order to use the leaf for a certain purpose, due to the prohibition of “Tikkun Keli”. This prohibition applies even if the branch is detached from the tree.
May one remove dates from its stalk?
May one remove bananas, grapes and other fruits from their vine?
One may do so immediately prior to eating. It is however forbidden to do so for later use.
 Peri Megadim M”Z 336/8 brought in Ketzos Hashulchan 142 footnote 12
From the letter of the law so long as the branch can no longer live, one may detach fruits from it even if it is still slightly attached. However since people are not expert in this matter it is only permitted to do so if the branch has become completely detached. [ibid]
 Ketzos Hashulchan 126/5
 Peri Megadim A”A 320
 This is forbidden due to the Mifarek prohibition, as it is the common way to remove the stalk of dates from the tree and then cut the dates off the stalks and send them to storage. This is exactly similar to threshing. Others however argue that threshing only applies to fruits which are concealed within their stalks, as are grains, and not by revealed fruits. [see Ketzos Hashulchan ibid footnote 9]
 Ketzos Hashulchan 126 footnote 10-11; Shabbos Kihalacha Vol. 2 p. 316;
 Some Poskim [Peri Megadim A”A 320] rule that the “Dosh” “Threshing” prohibition applies equally to items which are removed from their stalk even if the fruits are revealed. Meaning that not only by concealed fruits such as legumes in their pods does the prohibition apply, but also by all foods which are attached to a stalk. Nevertheless even according to them it is only forbidden when done in order to store, or eat later on, as is the form of Melacha of threshing that it is done for storage purposes. If however it is done to eat right away then it is allowed according to all, as this is not like Dash at all. Thus one may remove bananas from their stalk, and grapes from their vines to eat right away. However this is only by fruits which it is common to eat the food soon after removing it from its stalk. If however it is common to store the fruits after removing them from their stalk then they have the same law as does removing dates, in which some rule it is forbidden even to do in order to eat right away due to the threshing prohibition. [see Ketzos Hashulchan ibid footnote 9-10]