Or, more specifically, what is/are the reason(s) for not saying tachanun on various occasions and how does Tu b'Shvat fit in.
(I'm looking for classic reasons, not 'it's got to be somebody's birthday today')
The Shulchan Aruch (OC 131:6) brings down the minhag not to say tachanun on Tu Beshvat. It is quoted earlier in the Beit Yosef in the name of the Rokeach (Siman 312 in the Rokeach according to the Beit Yosef but I don't see it there). The Beit Yosef (quoting the missing Rokeach) says the reason is because it is the Rosh Hashana for trees. The Gra on the Shulchan Aruch there (sk 13) explains that all four Roshei Shana are Yamim Tovim (וכמו כל ד' ר"ה שהן י"ט). The Magen Avraham there (sk 16) quoting the Tikkun Yissachar Daf 25 (bio) explains that the minhag in Ashkenaz is to celebrate Tu Beshvat by eating lots of tree-fruits.
Rabbi Eliezer ben Shimshon (brought in Mordochai Mo'ed 701) was asked about a community which had declared a set of fast days and one ended up falling on Tu Beshvat. The community wanted to know if they needed to push off the fast. He ruled that they should push off the fast, because the Mishna (Rosh haShana 1:1) lists all the Rashei Shanah together, which he understands to be implying that their status viz-a-viz fasting is equivalent. Perhaps this opinion is the original source.
T"u ViShvat Sameach!
Tu b'shvat is one of the four Roshei Hashanah brought in the first mishna of maseches Rosh Hashanah. These days generally have a status of Yom tov, or at least a semi-moed, and therevore are days of some type of simcha, and we do not say tachanun.