The obligation to take a lulav on succot comes from the pasuk ולקחתם לכם ביום הראשון הדר כפות תמרים וענף עץ עבות וערבי נחל And you shall take on the first day the fruit of goodly trees, branches of palm-trees, and boughs of thick trees, and willows of the brook. 
- The spine of the Lulav must be at least 4 Tefachim.
- The lulav must have leaves that cover the spine. 
Acquiring the Lulav
- One should buy a lulav with a hechsher (certification that it is from a reliable source) in order to be sure that it came from a date palm tree and not a Canary palm.
- One can acquire a Lulav from one's friend as a gift even if there's a condition that says one must return it afterwards, as long as one actually does return it afterwards.
Leaves Separated from the Spine
- It is preferable to buy a lulav with leaves that are not separated from the spine. If, however, the leaves are somewhat separated from the spine, the lulav is acceptable as long as the leaves are not drooping downwards. 
- If the majority of the middle leaf is split, according to Ashkenazim the lulav is invalid, while some Sephardic poskim are lenient.
- It is preferable to get a lulav that is completely closed, if it is easy to get one.
- Ideally, one should have a completely straight lulav. If the spine of the lulav is bent, it is still acceptable unless it is as bent as a sickle. One should avoid using a lulav where the majority or all of the top leaves are bent like a reed.
- A dried out lulav is invalid. 
- If the uppermost tiyomet is dried out there is a discussion if it is still valid. 
- Vayikra 23:40 and Gemara Sukkah 34b
- Rabbi Yochanan’s opinion in the Gemara (Sukkah 32b) is that the spine of the lulav must be a minimum of 4 Tefachim. Based on Tosfot (32b s.v. Tzei), the Tur (650:1) holds that the lulav need be only 13.3 etzba’ot (rather than 16) since Rabbi Tarfon allowed using the smaller Tefachim to measure the lulav. The Ramban differs slightly and requires 14 etzba’ot. The Beit Yosef (650:1), however, infers from the fact that the Rif and Rambam do not cite Rabbi Tarfon that they hold that 16 etzba’ot are required. The Shulchan Aruch (650:1) quotes all three opinions and seems to side with the Tur. The Rama, though, writes that the minhag is to follow the Rambam. Chazon Ovadia (p. 362) writes that it is preferable to be strict for the opinion of the Rambam.
- There is great dispute about the length of a tefach: Rav Avraham Chaim Na’eh holds it is 3.2 inches, Rav Moshe Feinstein and the Aruch Hashulchan hold it is 3.6 inches, and the Chazon Ish holds it is 3.8 inches. See Rabbi Chaim Jachter’s essay in Kol Torah (Parshat Ki Tavo Vol. 13 Num. 2).
- The Gemara (Sukkah 32a) explains that a lulav whose leaves do not cover its spine is invalid. This is codified by the Rambam (8:4), Tur, and Shulchan Aruch (645:4). The Beiur Halacha (645:1 s.v. V’adayin) cites a view that a lulav is invalid only if the majority of the spine is uncovered, but he leaves the matter unresolved. The Chazon Ish (146:21), however, sides with those who require the entire spine to be covered.
- Rav Moshe Feinstein (Igrot Moshe O.C. 4:123) holds that a branch of a Canary palm tree, which does not produce edible dates, is not considered a lulav. Rav Shlomo Zalman Auerbach (cited in Halichot Shlomo 10:9), however, argues that the date palm and Canary palm are of the same species. Rav Hershel Schachter (“Halachos of Daled Minim,” min. 15) favors the former approach. The Halachos of the Four Species (p. 100) details how to distinguish a Canary palm branch from the regular date palm branch; one indication is that the spine of the Canary palm is very flexible. For more sources about a Canary Lulav see: Tzitz Eliezer 8:22, Kashrut Arbat HaMinim p. 170, and Yechave Daat 1:67. See Sefer Sukkat David (p. 52) who writes that if a Lulav came from a Lulav tree which does not grow fruits it is acceptable for the mitzvah of Lulav.
- Shulchan Aruch 558:4 writes that one can acquire the lulav through a gift with a condition that it must be return afterwards if one actually does return it afterwards. Mishna Brurah 558:12 explains that when one gives the Lulav to one's friend as a gift it is a complete acquisition and returning it later is only a completion of a condition that is imposed upon the gift. See, however, Shulchan Aruch 448:3.
- The Mishna (Sukkah 29b) states that a lulav with parted leaves is valid, while a lulav with leaves split completely is invalid. Rashi (29b s.v. Nifratz, as understood by Tosfot) explains that the lulav is invalid only if the leaves are actually detached from the spine. Tosfot, however, question Rashi’s approach. In a sense similar to Rashi, the Rif (15a) and Rambam (8:3) explain that the Mishna invalidates a lulav that has leaves split to the point that the leaves drooped downwards. The Shulchan Aruch (645:1) quotes the opinion of Rambam and Rif; the Beit Yosef quotes from the Ran that everyone would agree that by a lulav where the leaves fell off, it would be invalid, albeit for other reasons.
- The Maggid Mishneh (Hilchot Lulav 8:3) cites the view of the Geonim that although a lulav with parted leaves is acceptable, it is preferable to get a lulav with leaves that are not separated from the spine. The Rama (645:1) codifies this position and adds that it is preferable to have a lulav with leaves that are completely bound straight with the spine. The Taz (645:1-2) strongly questions the Geonim and Rama and concludes that a lulav with slightly parted leaves is acceptable even l’chatchila. The Mishna Brurah (645:3) and Kaf Hachaim (645:7), though, cite several achronim who side with the Rama.
- The Gemara (Sukkah 32a) states that if the leaves of lulav are stiff like wood, the lulav is invalid. Rashi (s.v. Charut, explained by the Beit Yosef 645:2) explains that when the lulav is left on the tree for a long time, its leaves harden so much so that it is impossible to bind them to the spine. Although this Gemara is quoted by the Tur and Rama (645:2), the Rambam and Shulchan Aruch make no mention of it. See Pri Megadim (E”A 645:1), who explains how the Rambam understood the Gemara.
The second version of Rabbi Yehoshua ben Levi’s opinion (Sukkah 32a) is that a lulav with a split tiyomet is invalid. Tosafot Sukka 29b "Niktam" explains that this is because it is a lack of "hadar." However, according to Rabbeinu Yerucham cited by Magen Avraham 645:6 it is an invalidation because it is "chaser." Magen Avraham agrees with the latter opinion, and therefore, MB 645:17 says that this invalidation wouldn't apply after the 1st day of Sukkot.
- Rashi (s.v. Tiyomet) explains that the tiyomet is the highest middle two leaves extending directly out of the top of the spine. Tosfot (Bava Kama 96b) cite the Geonim as agreeing with Rashi but note that according to this explanation it would be almost impossible to find a valid lulav, as almost every lulav in Tosfot’s area grew with one middle leaf rather than two. Tosfot, though, explain that even Rashi and the Geonim would agree that a lulav that did not grow with two middle leaves is valid; the only problem referred to in the Gemara is if a lulav grew with two middle leaves that subsequently split.
- The Terumat Hadeshen (Responsa 96) cites one version of Rashi (Bava Kama 96a s.v. Hatiyomet) that the tiyomet is the single tallest middle leaf. The Terumat Hadeshen writes that the minhag is in accordance with this explanation. The Rama 645:3 writes that the Ashkenazic minhag follows the Terumat Hadeshen. Mishna Brurah (645:16) adds that one should not use a lulav if the majority of the middle leaf is split. The Gra 645:11 says even if a minority is split based on an opinion quoted in the Ritva Sukka 31b, one should be stringent. Rav Shlomo Zalman Auerbach (Halichot Shlomo 10:1) explained that that minhag is not concerned for the opinion of the Geonim who required a “double” tiyomet. Chazon Ish 145 as well as Aruch Hashulchan 645:10 say that if necessary, one could be lenient even if most of it is split and recite a beracha on it.
- The Rif (15a), Ramban there, and Rambam (8:4) explain that every leaf is really doubled over, and the point at which these sides meet is called the tiyomet. The Rosh (3:6) explains that according to the Rif, if the majority of the length of the majority of the leaves split, the lulav would be invalid. The Shulchan Aruch 645:3 codifies the opinion of the Rif. Chazon Ovadia (Sukkot p. 297-8) codifies the opinion of the Rama even for Sephardim, while the Ish Matzliach (on Rama 645:3) rules that if there is no other lulav, Sephardim may rely on the Rambam.
- Interestingly, the Maamar Mordechai (645:4) writes that one should not check the middle leaf too forcefully, as it may split in the process; if it does not look split upon glancing at it, it is valid. This view is quoted by the Kaf Hachaim (645:24).
- Rama 645:3 writes that it is preferable to get a lulav that is completely closed to be strict for the opinion of those who hold that if the middle leaf is even partially split, the lulav is invalid. Chazon Ovadia (p. 300) agrees. Mishna Brurah 645:18-9 writes that if only the minority of the middle leaf is split one need not be strict unless another lulav is available to him.
- The Gemara (32a) states that a bent lulav is invalid if it is as bent (“akum”) like a sickle. The Gemara adds that if the leaves of the lulav are bent backwards towards the spine, it is nevertheless valid, since that is the normal way for it to grow. The Rambam (8:3) and Shulchan Aruch (645:8) codify this Gemara.
- The Gemara also says that a lulav whose leaves are bent (“kafuf”) is invalid. The Rosh (Responsa 24:10) explains that it is invalid only if the leaves and the spine both are bent; if the leaves alone are bent, it is valid. In fact, the Rosh mentions that he preferred a lulav with the middle leaf bent in order to ensure that it was not split. Tur and Shulchan Aruch (645:9) rule in accordance with the Rosh. The Mishna Brurah (645:40) limits the Rosh’s leniency to where only the middle leaf is bent and not if majority of the leaves are bent. The Chida (Machzik Bracha 645:4) writes that ideally the lulav should be completely straight.
- Mishna Sukka 3:1 (29a), Shulchan Aruch 645:5.
- Rosh Sukka 3:1 quotes the Raavad that when it loses its greenness it is considered dried out and this is the opinion of the Shulchan Aruch
- The Rama 645:5 is more lenient and says it is not considered dried out unless you would rub your fingernails on it and it would crumble.
- Mishna Brurah 645:22 is stringent based on the opinion of the Raavad as is the Gra 645:22, while the Chazon Ish 145:11 is lenient.