What should I look for in a Hunting Horn?
What To Look For In A Hunting Horn
Stats
The two most important things to look at when choosing a Hunting Horn (HH) are it's Physical Attack and it's Song List. Sharpness is also important, but since you get ESP (never bounce) with the self-improvement song it's not as important on other weapons. You should factor in affinity on a case by case basis, with every -/+ 10% affinity resulting in a net difference of -/+ 2.5% physical attack.
You can completely ignore the Elemental Attack of a Hunting Horn, because it contributes very little to the overall damage of a Hunting Horn. You get much more bang for your back from RAW damage than from Elemental damage when using a Hunting Horn. Even if you have situations where you would get a little more damage by choosing the Element the monster is weakest to you'd just be better off choosing a good song list anyways. In the end, your song list is far more important than any small increase in damage you might get from using the proper element.
Songs
There are a large number of songs in the game, and making sure you bring along the best song for the fight is very important. Some songs are great in all situations, others are situational, and some are just useless.
How Songs Work
Before I go into which songs to choose I should explain a bit about how songs work, since a lot of Hunting Horn players don't seem to fully understand this. First off, you should know that your self improvement song (Movement Speed Up + Attack Deflection Prevention) only affects you. It doesn't affect anyone else in your party.
The second thing you need to know is how song duration works. The first time you play a song it's applied to all players, and is set at the maximum duration for the song. The song duration varies from song to song, but it's usually between 1-2 minutes. The next time you play a song the duration of the song is increased (duration amount depends on song), it's not reset to the maximum duration. For example, if a songs max duration is 2:00 and playing it again will extend the duration by 0:30 then playing the song when there is a 1:10 left will result in there being 1:40 left on the song, and playing it again will result in there being 2:00 left on the song.
Last, but not least, you need to know about double playing. Almost all songs have a weaker and stronger version of the song. Playing a song the first time results in the weaker version being played, which is the one listed in your song list. The second time you play the song (that is, if you play the song again whilst players are already under its effects) you'll play a stronger version of the song, which offers some additional benefit. You should also know that it's possible to play a song twice for every time you queue it up, you just need to press the recital button twice (just like you would for a regular weapon combo).
All-Around Good Songs
Attack Up: An excellent song, and you should make sure you have a HH with Attack Up (L). The way the Attack Up songs work is that Attack Up (S) gives players +10% to Physical Attack, and Attack Up (L) gives players +15% to Physical Attack. The Attack Up Bonus adds another +5% to the physical attack, resulting in Attack Up (S) giving +10%, Attack Up (S) Bonus and Attack Up (L) giving +15%, and Attack Up (L) Bonus giving +20%.
Defense Up: Not as good as Attack Up, but still good. I'm not sure what the exact boost is, but more defense is always helpful. The reason I consider this inferior to Attack Up is just because the best defense is to just not get hit to begin with, so skilled players will get more use out of extra attack than extra defense.
Negate Stamina Use: Arguably the best song the HH user has, and easily the most widely loved song by Dual Sword users. Whilst Negate Stamina Use is in effect players won't lose stamina, and thus players using weapons which consume stamina no longer need to worry about stamina consumption. If you're playing with a group of people making use of stamina consuming weapons than I'd definitely suggest taking this song along, but it's good in all situations. Not needing to consume stamina while dodging is great for keeping yourself alive, and also makes it easier to travel between areas.
Hearing Protection: This is a tad situational, in that if you're fighting a monster which doesn't roar it won't be of any help, but most monsters do roar so it's great in almost all situations. The free attacks you'll get against monsters which roar a lot will make the battle go a lot faster, so it's a great help. Hearing Protection (S) and Hearing Protection (L) are equivalent to Earplugs and High-Grade Earplugs respectively.
Situational Songs
Negate Tremor Resistance: This is really helpful when you're fighting monsters which can cause tremors, like a Diablos or Deviljho. Not only does it make you less vulnerable to attack, but will also offers additional chances for attack.
Wind Pressure Negated: Flying monsters usually inflict Wind Pressure, and Wind Pressure Negated will negate the effects of it for you and your party. Wind Pressure usually doesn't make you more vulnerable, but it does give you less chances to attack, so negating it is a good help for the party.
Songs Not Worth Using
Healing Songs: Healing Songs heal for a really small amount, and always have a chance to fail and play a weaker version of the song. In fact you're more likely to play a weaker version of the song than you are to play the regular version. The time it takes to pull off the combo means it's never going to be useful in any sort of "Oh shit!" situation to save someone from almost certain death, and the heal amount is so small it probably wouldn't matter anyways. You'd be much better off just putting your horn away and eating a lifepowder to help your party members in "Oh shit!" situations.
Sonic Wave: Unless you keep this song queued up and wait around to play it then you'll rarely if ever going to actually get the Sonic Wave off in time. Just bring Sonic Bombs against monsters you want to use Sonic Waves against, you'll have a much higher success rate.
Suggested G-Rank Hunting Horns
I'm not going to waste time listing good horns for every rank, but I'll list a few good Horn to use once you get to G-Rank.
Nether Sinphony: My favourite HH in the game. It has an excellent song list and good attack. It has a huge bar of white sharpness and a some Purple Sharpness with Sharpness +1. It's notable for being the only G-Rank HH to give you Attack Up (S) and Negate Stamina Use, which is why it's my preferred HH.
Archgigas Gaita: Great attack and a good song combination, offering Hearing Protection and Attack Up (L). It's lacking in the sharpness department,, but with Sharpness +1 or Razor Sharp you won't need to worry that much about Sharpening, and bouncing won't be a problem because of your Self-Improvement song which offers ESP.
Sandsoother: Excellent attack, and even better once you factor in Attack Up (L). After you account for AuL, Sharpness (with +1) and the -25% affinity the HH has an effective Physical Attack of 2028, which is the highest possible Attack of any HH.
I agree, Nether Sinphony is my favorite by far. However, I'd like to point out both Stygian Tristitia and The Apocalypso.
Stygian Tristitia has decent attack at 1150 raw/320 dragon, and with Attack Up (Large) and a quick, 3 note, easily spammable Melody Durations Extended, it's a breeze keeping your mobility, fencing and attack up melodies in play.
As for The Apocalypso, it's raw damage falls a tad short at just 966, but the 600 elemental fire damage makes up for it, especially when used in conjunction with the Elemental Attack Boost melody (This horn absolutely wrecks monsters like Agnaktor and Glacial Agnaktor). Add to that Blight Resistance Boost, Blights Negated, and a single melody for Hearing (Small) & Windproof, as well as natural purple sharpness, you've got yourself one decent hunting horn, albeit best used on monsters with elemental weakness to fire.