Since the groups system has no better mechanism for collecting recommendations which aren't stories, here are the songs I recommend as a forum post.

Unlike stories (where my "read it later" list grows despite my best efforts), I do welcome suggestions here:
1. PM suggestions to me
2. I'll listen to them when I have time.
3. If I like them, they'll be added to this post.

If possible, please explain why you think they're awesome. It is possible for songs that don't fit my musical tastes to wind up on this list but, given how busy I often am, I may need help with what to pay attention to. (If I like a song, I'll play it on repeat and, sooner or later, I'll notice.)

Also, just because an artist is listed here doesn't mean I have heard everything by them so suggest freely. (As of a few months ago, I've heard everything I could find by ponyphonic, StormWolf, and WoodenToaster but I'm quite haphazard in how I listen and I don't think I'm subscribed to updates on them.)

General:
The Mirror Lies by Freewave & SylverLining
Lady Rarity (feat. DJ Shamrock)
I Might Be A Brony Now by @EdgeOfEverfree
Don't You Worry Now Scootaloo by Coconeru
So Much Left To Know by MandoPony

Ponyphonic (artist):
Winter Wrap-Up (cover)
Shy Heart (Added 2016-12-06)
Harmony Ascendant (could also be a distant "The Moon Rises" prequel) (Added 2016-12-06)
Feed Me with Your Heart (Added 2018-10-31)

The Moon Rises (series):
The Moon Rises by ponyphonic (extended & animated)
Lullaby for a Princess by ponyphonic
Luna's Reply by MsIkarishipper

StormWolf (artist):
Perfect Picture
Shadow of a Flower
The Beginning of Order
Nightmare Night (orchestral cover)

Eurobeat Brony (artist):
Luna (DREAM MODE)
Discord (EuroChaos Mix)

WoodenToaster (artist):
Nightmare Night (with Mic The Microphone)
Beyond Her Garden
Fruits of Her Labour (with StormWolf)
Rainbow Factory
Love Me Cheerilee (with The Living Tombstone)

Runners-up (any artist):
BBBFF (StormWolf Remix)
She's A Pony by General Mumble (Remixed by WoodenToaster)
Lesson Learnt by StormWolf
Diffraction [Rainbow & Rooted] by StormWolf
Pinkie's Lie by WoodenToaster
《FULL》 Bad Harmony 《Bad Apple→MLP:FiM》
Avast Fluttershy's Tushy by WoodenToaster (Yay-sampled cover of Avast Your Ass)

...and, until I have enough to merit their own thread, animatics:
A Story Told (from the Count of Monte Cristo musical)

--== ADDED 2016-07-29: ==--
Nightmare Night 8 Bit Style Woodentoaster by Lukspups
Tsyolin Befriends WoodenToaster - WoodenOverture (medley)

Gypsy Bard (and covers):
Original version from Friendship is Witchcraft 7
Dreamchan's soulful guitar+vocals cover
The Living Tombstone's techno remix with olibacon's impressive visuals
BassBeastJD's orchestral instrumental cover

--== ADDED 2017-07-02: ==--
Why Am I Pinkie Pie? (mirror) by DnBeanie (Inspired by a fic I still need to move to this group from the old list on my profile page)

Evening Star (artist):
Friendship is Magic (Orchestral Arrangement)
Octavia

--== ADDED 2018-02-20: ==--
Nightmare Night cover by SlyphStorm

--== ADDED 2019-05-05: ==--
Octavia's Overture cover by SlyphStorm

--== ADDED 2019-12-31: ==--
Who Knows by 4everfreebrony

--== ADDED 2020-03-21: ==--
Daniel Ingram - Luna's Future (feat. Aloma Steele) [Jyc Row 2017 edit ~ 3k subs special]

--== ADDED 2020-05-14: ==--
Neverending Strife - SlyphStorm (redux)