Gemara

6b: There was a bent Mavoy in Neharda'a. They were stringent like Rav, who considers it like a Mavoy Mefulash, and like Shmuel, who requires doors.

Question: Rav never requires doors, and Shmuel considers it like a Mavoy Sasum! It is improper to follow the stringencies of both!

(Beraisa): One may choose to follow Beis Shamai, or Beis Hillel;

If one follows the leniencies of both, he is wicked;

It says about one who follows the stringencies of both "veha'Kesil ba'Choshech Yelech" (a fool walks in darkness).

Rather, one consistently follows Beis Shamai, or consistently follows Beis Hillel. (This was before a voice from Heaven announced that the Halachah follows Beis Hillel, or it is like R. Yehoshua, who does not heed such voices.)

Answer (Rav Shizbi): One should not follow stringencies of different Chachamim only if they are contradictory, like the following:

(Mishnah - Beis Shamai): If two vertebrae are missing from the spine, and the width of a drill-screw is missing from the skull, they have no Tum'as Ohel;

Beis Hillel say, if one vertebra is missing from the spine, and enough was Chaser (missing) from the skull to kill a live person, they have no Tum'as Ohel.

(Rav Yehudah): The same amount missing makes an animal Tereifah. (If the smaller Shi'ur of Chisaron makes a Terefah, one cannot say that there is Tum'as Ohel until the larger Shi'ur is missing.)

When the stringencies do not contradict each other, one may follow both.

Question (Rav Mesharshiya): Even when they contradict each other, one may follow both stringencies!

(Beraisa): R. Akiva once separated [Ma'aser Oni] like Beis Shamai [who say that Rosh Chodesh Shevat begins the new year for trees], and Ma'aser Sheni like Beis Hillel [who hold that the new year begins on Shevat 15].

Answer: No, R. Akiva was unsure whether Beis Hillel hold that the new year is the first or 15th of Shevat, therefore he was stringent.

Berachos 26b (Beraisa): One may pray Minchah until evening;

R. Yehudah permits [only] until Plag ha'Minchah.

27a: Since the Halachah was not fixed like either opinion, one may follow either one.

Chulin 43b: A case occurred in which the Shechitah started in the Turbatz Veshet, and finished in the Veshet proper.

Rava: I will apply the stringencies of Rav and Shmuel to rule that it is Tereifah;

Rav holds that a cut of any size in the Turbatz Veshet makes it a Tereifah.

Question: Rav holds that also Turbatz Veshet is a valid place for Shechitah!

Answer: Rava rules like Shmuel, that it is not.

Question: Shmuel holds that only cutting the majority of Turbatz Veshet makes it a Tereifah!

Answer: Regarding this, Rava rules like Rav.

Question (against Rava - Mar brei d'Ravina): A Beraisa teaches that one may not rule like contradictory stringencies!

Answer (Rav Tavos): Rava ruled entirely like Rav.

Really, Rav holds that Turbatz Veshet is invalid for Shechitah. Even so, if it is cut at all, it is Tereifah.

Beitzah 14a (Mishnah - Beis Hillel): [On Yom Tov] one may mash spices with a stone masher, and salt with a wooden masher.

Question: Why do we require a Shinuy for salt?

Answer #1 (Rav Huna or Rav Chisda): All cooked foods need salt. (He should have prepared before Yom Tov);

Answer #2 (the other of Rav Huna and Rav Chisda): Salt's taste is not dulled if it is mashed beforehand, unlike other spices.

They argue about one who knew which spices he will need, or saffron. (It does not dull.)

(Shmuel): One may mash all spices normally.

Rishonim

Rambam (Hilchos Tefilah 3:4): One may pray [Minchah] until Shki'as ha'Chamah.

Kesef Mishneh: The Rambam holds like the Rosh brought from a Gaon (below).

Rosh (Berachos 1:1): R. Tam says that sometimes we pray Minchah after Plag ha'Minchah, like Chachamim. Regarding Keri'as Shema, we consider this time to be night, like R. Yehudah. These are contradictory leniencies! We can say that regarding Tefilah, they were lenient. I say that there is no proof from Minchah to Keri'as Shema.

Rosh (4:3): A Gaon wrote that one who prays Ma'ariv early may pray Minchah only until Plag ha'Minchah, like R. Yehudah. The Halachah follows R. Yehudah or Chachamim, but not both of them! One cannot do once like R. Yehudah and once like Chachamim.

R. Yonah (18b DH d'Avad): We must do totally like Chachamim. However, b'Di'eved if one prayed after Plag ha'Minchah with intent to be Yotzei Ma'ariv, he was Yotzei. We do not obligate him to pray again.

Rambam (Hilchos Yom Tov 3:12): One may mash spices normally, for if he would mash them from before Yom Tov, the taste would dull.

Lechem Mishneh: The Beis Yosef asked that according to this opinion, saffron is forbidden. I say that we hold like Shmuel, that everything mashed may be mashed normally. He taught Stam, i.e. even saffron. Even salt needs only a small Shinuy. Since it is a small matter, it is proper to be lenient, since it is a mid'Rabanan law. We adopt the leniencies of both opinions.

Tosfos (Beitzah 14a): The Ri is stringent for both opinions. Even though we follow the lenient opinion for mid'Rabanan laws, each opinion has a leniency and stringency.

Rashba (Chulin 44a DH Ela): Our Gemara connotes that when two Tana'im or Amora'im argue, and the Halachah was not decided like either of them, one may follow the leniencies and stringencies of one, or the leniencies and stringencies of the other. Eruvin 7a connotes like this. This is when each opinion has a leniency and stringency, like the argument about a spine and skull or a bent Mavoy. If one opinion is stringent and the other is lenient, we follow the bigger Chacham. If they are equal, for Torah laws we follow the stringent opinion, and for mid'Rabanan laws we follow the lenient opinion (Avodah Zarah 7a).

Rashi (Rosh Hashanah 14b): When there is a leniency and a stringency of each opinion, one who adopts both leniencies is a Rasha, for the leniencies contradict each other. One who adopts both stringencies is a fool, for he does not know whom to rely on. Regarding two arguments that do not depend on each other, one is not a Rasha or fool [if he adopts both leniencies or stringencies]. He holds like Beis Hillel regarding one, and regarding Beis Shamai regarding the other!

Teshuvas Mizrachi (56): I hold like the Rambam's leniency, but not his stringency to forbid [wine above the spigot, where the Nochri touched]. I am lenient like Rashi, who forbids only drinking, but not Hana'ah. This is because the leniencies do not contradict each other.

Poskim

Shulchan Aruch (OC 233:1): One must consistently follow one opinion. If he follows Rabanan and prays Minchah until night, he may not pray Ma'ariv after Plag ha'Minchah. If prays Ma'ariv from Plag ha'Minchah like R. Yehudah, he may not pray Minchah at that time. Nowadays we pray Minchah until night, so one may not pray Ma'ariv before Shki'ah. B'Di'eved, if one prayed Ma'ariv after Plag ha'Minchah, he was Yotzei. One may do so in pressed circumstances.

Beis Yosef (DH u'Mah she'Chosav v'Hu): Ohel Mo'ed brings from Rav Hai Gaon that one must consistently follow one opinion. Ba'al ha'Hashlamah says that nowadays, due to pressed circumstances, people sometimes follow R. Yehudah, and sometimes Rabanan. We find that one who rises early to travel may recite Shema earlier than the normal time. R. Yonah is stringent. Nowadays, people pray Ma'ariv after Plag ha'Minchah, and pray Ma'ariv in the same time period, even though the leniencies contradict each other. They rely on R. Tam. Also the Rashba says that we are lenient about Tefilah. The Rosh rejected R. Tam; if one prays Ma'ariv with a Tzibur before Tzeis ha'Kochavim, he may not pray Minchah after Plag ha'Minchah, since he considers this night regarding Ma'ariv. Even so, people rely on R. Tam.

Kaf ha'Chayim (9): A woman need not be concerned about contradictory leniencies. Since she is busy with needs of the house, it is like pressed circumstances for her.

iv. Kaf ha'Chayim (10): Olas Tamid says that a Chasid should pray Minchah before Plag ha'Minchah, and Ma'ariv after Tzeis ha'Kochavim, to fulfill both opinions.

Rema: Here we pray Ma'ariv from Plag ha'Minchah, so one may not pray Minchah afterwards. B'Di'eved, or in pressed circumstances, one was Yotzei if he prayed Minchah until night, i.e. until Tzeis.

i. Darchei Moshe ha'Aruch (1): Nowadays, the custom is to pray Minchah and Ma'ariv next to each other. We hold like Chachamim; we pray Ma'ariv before night due to the difficulty of gathering people a second time. If one did not pray Ma'ariv with the Tzibur, he should wait until night.

ii. Gra (DH v'Im): Even though the Rosh disagreed with R. Tam, b'Di'eved or in pressed circumstances one may rely on R. Tam.

iii. Mishnah Berurah (11): In pressed circumstances, one who normally prays Minchah after Plag ha'Minchah may pray Ma'ariv at this time, if today he prayed Minchah beforehand. One may not contradict himself on one day! However, if a Tzibur prayed Minchah, and if they would go home it would be hard to gather them again, and Tefilah b'Tzibur would be totally Batel, Acharonim permitted Ma'ariv close to Minchah.

v. Kaf ha'Chayim (12): The Ari Zal would pray Minchah close to sunset, and Ma'ariv shortly after, so that all three Tefilos are during the day. He relied on R. Tam.

Shulchan Aruch (504:1): One may mash spices normally, for if he would mash them from before Yom Tov, the taste would dull.

Rema: However, the custom is to make a small Shinuy. It is proper to rule like this.

Magen Avraham (1:4): The Shinuy reminds him not to mash more than needed for Yom Tov. The Yam Shel Shlomo requires a Shinuy only for saffron, or if he knew that he will need these spices.

Gra (DH Mihu): Tosfos is stringent, even though it is mid'Rabanan.

Pri Megadim (introduction to OC, Seder ha'Nish'al, 3:11): We can rule like leniencies of different Poskim, as long as they do not contradict each other.

Teshuvas Radach (1, Os 28): One may not do like the stringencies of different opinions. This is only if they contradict each other. One may not do like the leniencies of different opinions in any case. One must be concerned lest he is called a Rasha.