Gemara

(Mishnah - R. Meir): One who is missing a foot may go out with a Kav (false foot. It houses what remains of his leg, it is a shoe for him);

R. Yosi forbids. (It is not a Tachshit.)

(Abaye): If [a Kav] has a receptacle for tufts it is Mekabel Tum'as Mes, but not Tum'as Medras. (He does not usually lean on it.)

(Rava): It is Mekabel even Tum'as Medras.

Support (Abaye, for himself - Beraisa): A walking stick for an elder is totally Tahor [even though he leans on it sometimes].

Rejection (Rava): The stick is merely to straighten his steps. (He does not lean on it at all);

The Kav is made to lean on, and he leans on it.

Beitzah 25b (Beraisa): A blind person may not go out with his stick. One may not go out with a chair, both men and women.

Rishonim

Rosh (6:17): Rashi says that an amputee may go out with a peg leg. It is known that such people cannot walk without sticks (crutches) that he holds in his hands. From here we learn that a Chiger (lame person) may go out with sticks, even though he holds them in his hands and they are not tied to him. R. Tam permitted one whose thigh sinews shriveled to go out with sticks on Shabbos. It seems that this is only if he cannot walk at all without them due to illness. If an elder walks in his house without sticks, and when he goes outside he leans on his stick due to weakness, he may not go out with it on Shabbos.

Rosh (Beitzah 3:5): Even though the Ri explains that permitted one whose thigh sinews shriveled to go out with sticks on Shabbos, this is only if he cannot walk at all without them. It is like his shoes. For a blind person it is merely to straighten his steps. It is a disgrace to Yom Tov.

Mordechai (359): All agree that a Chiger or one whose sinews shriveled may go out with sticks. An elder's stick is forbidden. It is Tahor, for it is only to straighten his steps. (He does not lean on it.)

Poskim

Shulchan Aruch (OC 301:17): A Chiger who cannot walk without a stick may go with it even if it is not tied to him.

Mishnah Berurah (63): If he cannot walk without it, it is like his shoe.

Bi'ur Halachah (DH she'Eino): If there is a nail at the end of the stick and he makes a hole where he walks, this is a Pesik Reishei d'Lo Nicha Lei (he is not pleased with the result). This requires investigation whether he is permitted. Mishbetzos Zahav (12) leans to permit.

Kaf ha'Chayim (102): If he cannot walk without it, there is no concern lest he come to carry it. If there is a nail at the end, it is not clear if he may walk where there is no [stone] floor. One should avoid this.

Shulchan Aruch (ibid.): If he can walk without it, and he takes it only to strengthen himself, it is forbidden.

Beis Yosef (DH u'Mah): The Rosh, Tosfos, the Mordechai, Ran, Magid Mishneh in the name of the Rashba, Hagahos Maimoniyos, Semag, Semak, Sefer ha'Terumah and R. Yerucham say so.

Taz (12): It seems that if he walks in a place where there is concern lest he fall, e.g. it rained and the place is sloped, or he walks on ice, he may go out with a stick, for he fears to walk there. We say that one may not carry someone on a "chair" of arms, but it was permitted for Yalsa (Rav Nachman's wife) because she was afraid. It is forbidden for straighten his steps.

Mishnah Berurah (65): Eliyahu Rabah says that this is not necessarily true. Also other Acharonim permit only where there is an Eruv.

Magen Avraham (27): Distinguished people go out with a stick in their hands. This is forbidden. Do not say that it is like a Tachshit (ornament). We do not find that one may carry a Tachshit in his hand. Even though R. Eliezer holds that [weapons] are a Tachshit, he said so only to exempt from a Chatas. It is forbidden to go out with them. Tosfos (62a DH veha'Tanya) connotes that the Isur to carry a Tachshit in one's hand is only due to Mar'is ha'Ayin. Also Ra'avan (349) says so. The Yerushalmi says that R. Gamliel carried a [gold] key in his hand [and his colleagues rebuked him]. We cannot say that it was truly in his hand. If so, the decree 'lest he remove it and show it off' would not apply. We must say like the Ran says, that his finger was inserted in it, like a ring. This shows that one may not carry a Tachshit in his hand. Also, it is made for a Tachshit and a use! Where there is an Eruv, he may go out with it, since he wears it only for honor. A regular person, who does not wear it for honor, is forbidden even where there is an Eruv, for it disgraces Shabbos, like it says in 522:1 (brought below). The Tur says that one may not go out with a chair unless many people need him. This is where there is an Eruv. This answers the question of the Beis Yosef (DH u'Mah she'Chasav v'Im, who asked how the Tur learns from a Heter on Yom Tov to permit on Shabbos).

Kaf ha'Chayim (104): Eliyahu Rabah leans to permit a stick for honor. Machazik Berachah was unsure about the Magen Avraham's Heter within an Eruv. Perhaps it is not considered a Tachshit. It seems ridiculous to permit to some people and forbid to others. Not everyone knows the laws. If a Rav rules that only distinguished people are permitted, the lowly will contend with the distinguished. Everyone will say 'I am distinguished.'

Gra (DH Aval): We learn this from Sa'if 18 (a blind person may not go out with a stick).

Sha'arei Teshuvah (17): The Noda bi'Yehudah (2:28) says that it is not clear that the Taz permits on ice when it is not sloped. Even though the Eliyahu Rabah said that the Taz is not necessarily correct, one who relies on him is not called Mechalel Shabbos. If there is an Eruv, one who normally carries is permitted. This is not only if it is beautiful. It is even for a simple wooden stick that is not a Tachshit.

Kaf ha'Chayim (108): The Radvaz says that people who go out with a stick when it is muddy, they are forbidden on Shabbos and Yom Tov even within a Eruv, due to Uvda d'Chol (a weekday action).

Mishnah Berurah (64): If an elder walk in his house without it, and when he goes outside he leans on it due to his weakness, in order to support himself, even though he wobbles, it is forbidden. The stick is a load for him, since in his house he walks without it. If he is so old that he cannot walk without it at all, it is permitted.

Bi'ur Halachah (DH Aval): If it is like a load, there is an Isur Torah in Reshus ha'Rabim. The Pri Megadim initially said so, then concluded like the Levush that it is only a decree lest one come to carry it four Amos in Reshus ha'Rabim [without leaning on it]. Rava concluded that an elder's stick is not to lean on at all. The Mordechai brought a proof for his law from an elder's stick. R. Yerucham holds that the stick is not considered a shoe or Tachshit. Or Zaru'a explicitly says that it is like a load.

Rema: A Choleh who recovered is like a Chiger.

Mishnah Berurah (67): This means that it depends on whether or not he can walk without a stick.

Kaf ha'Chayim (106): The Beis Yosef (522) says that this is when he cannot walk without it. If not, he is like a blind person, for whom it is merely to straighten his steps (so it is forbidden - Sa'if 18). Shirei Keneses ha'Gedolah says that since the Mechaber did not bring this in Hilchos Shabbos, only in Hilchos Yom Tov, and the Rema brought it in Hilchos Shabbos, this shows that the Mechaber holds that the Roke'ach permits only on Yom Tov, when the Isur Hotza'ah is only mid'Rabanan.

Shulchan Aruch (522:1): A blind person may not go out with his stick. The law of a Chiger on Yom Tov is like on Shabbos.

Beis Yosef (DH Ein): Rashi explains that it is like a weekday matter for a blind person to go out with his stick. The Roke'ach (299) says that a Choleh who recovered may walk with a stick. It seems that this is only if he cannot walk without it. If not, it is forbidden like for a blind person, since it is only to straighten his steps.