A. Tearing an item in the process of barbecuing fish on Yom Tov:
Tearing a piece of paper to place under the fish: One who roasts fish over a grill is not to cut a piece of paper in order to soak it in water and then place it under the fish over the grill in order to prevent the fish from burning.
Breaking a piece of earthenware to place under the fish: Similarly one may not break a piece of earthenware in order to place it under the fish.
Breaking open a cane to place its sheath under the fish: Similarly one may not break open a cane in order to place its sheath under the fish.
Making the cane into a skewer: As well, one may not break open the cane to make it into a figure like skewer to roast with.
The reason for all the above restrictions is: because in all cases that one makes and fixes an item to be fit for a use, it is like he has fixed a vessel on Yom Tov.
B. Cutting a vine to fit a use:
Using a vine as a rope to help draw water with: A [detached] vine which has its head split like a fork and is [thus] fit to [use to] hang a bucket on and to [thus use to] draw [water] with [such as to place it down a well to draw water, using the vine as a rope to place it down and bring it back up], then even if one thought about this from before Shabbos to use the [vine] to [help] draw [water] with on Shabbos, then [nevertheless] it is forbidden to use it to draw with, unless the vine was tied to the bucket from before Shabbos.
The reason for this is: due to a decree that perhaps the vine will be too long for him, and one will cut it, being that it is soft and easy to be cut, and will thus end up [transgressing the prohibition of] fixing a vessel on Shabbos.
C. May one remove a reed from ones broom?
It is forbidden to remove a reed from a broom which is used to clean the house, being that through removing it he is fixing it for the use that he wishes to use it for, which is for hitting the children with, and [the law is that] any item that one fixes to be used for any use is included in the prohibition of fixing vessels. [Furthermore] even through a gentile it is forbidden to remove it, meaning [to even have him] detach it and remove it out from under the binding area of the broom, and it goes without saying that it is forbidden to [ask a gentile to] break the reed off from the broom, as by breaking it there is an additional prohibition [being transgressed which is] breaking a vessel, as will be explained in chapter 337 [Halacha 3]. However when one removes out an entire reed [from the broom] there is no [prohibition involved] of destroying a vessel, as this is similar to a vessel assembled by placing many pieces together, which does not contain [the prohibition of] destroying [a vessel] when taking it apart, unless the [attached pieces] were inserted [in their sockets] strongly and professionally.
It is forbidden to break a reed off a broom, or even detach a reed from its setting in a broom, in order to make a use of it. It is forbidden to do so even through a gentile.
D. Using a twig as a tooth pick:
A twig which is not animal food, even to take it in order to pick at his teeth is forbidden [due to it being Muktzah, and to cut a piece off is forbidden also due to the prohibition of “fixing a vessel”].
E. Plucking a leaf from a branch to use as a funnel:
Is forbidden. See below Halacha 5 “Tikkun Keli by foods”
F. Cutting a knot:
Cutting the sewing and knot of a tailor: [Furthermore] even if the collar had already been opened [after being made] but a professional [tailor] returned and sewed [the two sides of the collar] together in the way done by professionals, or if the professional tied it [in a way] that one is unable to untie it, then it is forbidden to cut the strings.
However this [prohibition to cut an un-openable knot] only refers to a knot tied by the professional prior to having finished making the clothing, being that then the cutting of the strings of this knot involves the [prohibition of doing a] “Finishing touch”, as this finalizes the work needed to be done to the clothing being that through doing so the clothing is now fit to be worn, while until this was done it was never yet fit to be worn, [and it is thus forbidden as] any [action] done which is the finishing stroke [of the making of the vessel] contains [the prohibition of] “Finishing Touch” as was explained in chapter 302 [Halacha 5]
Cutting the knot of the launderer: See “The laws of Cutting and Tearing” Prohibition #2 Halacha 3E for the full details of this subject. See summary below for a summary what is explained there!
Summary: Cutting the strings of a knot:
Anytime it is permitted to open a knot on Shabbos one is likewise permitted to cut it. However one may not be lenient to cut it in front of an ignoramus, and rather should do it privately. [As well in a case that the knot attaches an item to the ground then the rope may not be torn even if one is allowed to untie the knot being that doing so involves the destroying prohibition.]
Anytime that the knot is forbidden to be undone, it is likewise forbidden to be cut.
G. Cutting and tearing sewn threads:
See “The Laws of cutting and Tearing” Prohibition #1 Halacha 4!
H. Tearing the covering off of a bottle:
A person may tear the hide which [seals] the opening of the barrel of wine or other liquids if he is doing so for the need of Shabbos. He need not worry [about any tearing prohibition involved in doing so], because tearing a detached item is allowed to be done even initially. [However this is only allowed] as long as one does not intend to make something similar to a drainpipe out of the torn skin [i.e. a funnel] being that doing so is considered like fixing a vessel.
Is permitted to tear the leather covering of a bottle for a Shabbos need so long as one does not form it into a drain [and it is a single entity like a piece of leather as opposed to a cloth].
Why is tearing the cover off a bottle etc not considered Tikkun Keli, that one is fixing the barrel?
As a cover of a bottle is not considered a complete attachment, and hence is considered as if it is not on the bottle.
If the cover has letters or designs may it be torn?
One may only tear it open around the letters.
 Lit. a branch
 Meaning out from the area where all the reeds have been fixed into.
 When breaking it off for a use it contains two prohibitions, destroying and Tikkun Keli. When simply detaching it from its setting, then if loose, it only contains the Tikkun Keli prohibition.
 Michaber 322/4
 If done with one’s hand it is Rabbinically forbidden, and if done with a vessel is Biblically forbidden. [M”B 13]
 317/ 6
 Chapter 314 Halacha 19
 See “The Laws of Cutting and Tearing” Halacha 2
 See Chazon Ish Shabbos 51/11