Rosh Hashanah, Chapter 3, Mishnah 6. This mishnah deals with various things that can happen to a shofar - a crack, a hole, or a break - and with the question of whether it remains valid or becomes invalid.
"Shofar shenisdak vedibko - pasul" - a shofar that cracked and was glued back together is invalid:
The mishnah opens with a shofar that cracked. The Rishonim disagree about the definition of "cracked":
Rashi: It means that it broke into two completely separate pieces.
Tosafos: Even if it cracked on one side only, provided that the crack runs along its entire length.
And some understand: Even if it cracked only slightly, since in the course of blowing the shofar the crack is bound to widen considerably.
Sadly, this happens easily. There is the story of a five year old boy who, at the end of the second day of Rosh Hashanah, dropped his father's shofar, and his father was the ba'al tokea. The shofar cracked and became invalid, and the father had to obtain a new shofar - the very shofar that is still in use to this day.
The mishnah teaches that even if he glued the cracked shofar and restored it to its original state, it is invalid; gluing does not help.
It should be noted, however: if it cracked in such a way that the crack does not run along the side of the shofar but rather from top to bottom, then as long as enough remains that when he holds it in his hand it still looks like a shofar from both sides, it is valid.
"Dibek shivrei shofaros - pasul" - if he glued together broken pieces of shofars, it is invalid:
The mishnah rules that one who glues together various pieces of shofar has an invalid shofar. But this ruling applies specifically when none of the pieces has the required measure, meaning that no single piece is large enough to count as a valid shofar on its own. If one of the pieces does have the measure needed for blowing, that piece by itself suffices to make the shofar valid, and gluing the other pieces onto it does not necessarily invalidate it. And some hold that once the pieces have been glued together it no longer comes from a single shofar, and therefore it is invalid.
"Nikav usetamo" - it was pierced and he sealed it:
The mishnah discusses a shofar that was pierced, meaning that a hole formed in it, and he then sealed the hole:
"Im me'akev es hateki'ah - pasul" - if it interferes with the blowing, it is invalid: when the hole prevents the sound of the blowing from being heard properly, so that the shofar cannot be used, then even if he sealed it, it is invalid.
"Ve'im lav - kasher" - and if not, it is valid: if the hole that was there before it was sealed did not prevent the blowing from working properly, then even though he sealed it, the shofar is valid.
In summary: in this mishnah we learned the laws of a damaged shofar: a shofar that cracked and was glued is invalid, and the Rishonim disagree about what sort of crack invalidates it, whereas a crack running from top to bottom that still leaves the measure of a shofar is valid; one who glues together broken pieces of shofars has an invalid shofar, unless one of the pieces has the measure needed for blowing, and even in that case there are those who disagree; and a shofar that was pierced and then sealed depends entirely on whether the hole interferes with the blowing.