Halachos L'Kavod Shabbos Shira

1) Shabbos Parshas Beshalach is known as "Shabbos Shira".

It is customary to put out pieces of bread for the birds to eat. One of the many reasons  given for this custom is that in the desert the Mann fell for six days, with a double portion falling on Erev Shabbos to sustain the Jews through Shabbos.

The two evil troublemakers in the desert, Dasan and Aviram, tried to disprove Moshe Rabeinu's proclamation that no Mann will fall on Shabbos, and they took their Mann early Shabbos morning and scattered it all over the camp so the jews will wake up and see that in fact Mann did fall on Shabbos.

Hashem sent the birds to eat up every last crumb of the "planted Mann"  and when the Jews awoke, it was as Moshe had promised, and no Mann was to be found.

Thus as a reward for this, the birds are fed each year on Parshas Beshalach.

2) he Mogen Avraham (Orach Chaim Siman 324:7) decries this minhag, as the Halacha is that one may not feed animals that do not belong to him/her on Shabbos, and thus one may not put out food for the birds during Shabbos Shira (or any other Shabbos). (This is also how the Mishna Berura Siman 324:31 and the Shulchan Aruch HaRav 324:8 rule.)

Other Poskim justify the minhag to feed the birds on  this particular Shabbos  as we are doing this as a Mitzvah of Hakoras HaTov (See Aruch HaShulchan 324:3 and Da'as Torah 324:11)

[ Incidentally, the Mogen Avraham Siman 171:1, based on Rashi to Ta'anis 20b rules that one may never feed food that is worthy of human consumption to animals or birds, as doing so  is being ungrateful to Hashem for the food He has given us]

The best thing to do, if one wants to uphold this custom yet avoid feeding the birds on Shabbos, is to place the bread for the birds right before Shabbos begins. (And if one wants to adhere to  the ruling of the  Mogen Avraham in Siman 171, the bread that is placed for the birds should be stale and not good fresh bread that is still worthy of human consumption)

Another option brought in the Poskim is to have a minor (boy under 13, girl under 12) put out the bread, as it isn't a real Issur (Aino Shevus Gamur). (See Aishel Avraham (Butchatch) Siman 167: 6 Dibur Hamaschil Sham)

Halachos L'Kavod Tu B'Shvat

1) It is customary [for Ashkenazic Jews] to eat different varieties of fruits on Tu B'Shvat, as this day is the 'Rosh Hashanah" for fruit trees (meaning that the "year" regarding Ma'aser and Arlah is counted from this day) (Mogen Avraham Orach Chaim Siman 131:16)

2) Some people have the custom to eat an Esrog (or Esrog Jelly) on Tu B'Shvat (Likutei MaHarich)

It is also customary to daven on Tu B'Shvat for a beautiful, kosher Esrog on the following Sukkos (Bnei Yissochor Shvat 2:2)

 The Sefarim write that the words "Chamisha Asar B'Shvat" are the Gematria (numerical value) of the words "L'Kavein Al Esrog Lulav Hadas Aravah (to have in mind for Esrog, Lulav, Hadassim and Aravos)"