First, let's define what I mean by the things in the title:

Direction Targeted: This video displays what I mean by directiontargeted. I used Yasuo's Q at a location not above the target dummy, then flashed so the direction my cursor was relative to Yasuo would hit the target dummy and it hit the target dummy. For blink or dash abilities with a delay, this will be used for if they go in the direction of the cursor before the flash, as with skillshots, but it will also be included whether or not they go their full distance.

Location Targeted: This video displays what I mean by location targeted. I used Zed's Q on the target dummy then flashed after I had already cast the ability and it still went toward the target dummy. For blink or dash abilities with a delay, this is a pretty simple distinction. A location targeted, delayed blink or dash will always do its best to get to the targeted location.

Flash Direction: This video displays what I mean by flash direction. I used Ashe W toward the left of the screen and flashed toward the right, and Ashe's W came out toward the right.

In-Range: Applies only to location-targeted, area of effect abilities. This video displays what an in-range, location targeted, area of effect ability is. When I cast Miasma then flash out of its range, Miasma's area of effect lands at the maximum range.

Out-Range: Applies only to location-targeted, area of effect abilities. This video displays what an out-range, location targeted, area of effect ability is. When I cast Tibbers and then flash out of its range, it will still cast at the original cast location, regardless of the fact that it's now out of range.

Flashable: As seen in the five videos above, flashable means you can flash after casting the ability and it will come out from your new location. Point-blank area of effect abilities will only include whether or not they are flashable, since they do not include the ability to be targeted.

Unflashable: As such, you can assume this is the opposite, but just for example, I also recorded the prime example of this, Lee Sin Q in this video.

In the table below, if an ability does not allow you to flash during it, such as Blitzcrank's Q, the Flashable column will list "Cannot," and Targeting will simply be noted with a dash(-). If I do not have a champion to test them, I will list their Flashable status as "Unable to test."

Champion Skillshots (A-C)

Ability Flashable Targeting Location Delayed Distance
AatroxSquare Aatrox's Dark Flight Dark Flight Cannot - - -
AatroxSquare Aatrox's Blades of Torment Blades of Torment Flashable Location - -
AatroxSquare Aatrox's Massacre Massacre Flashable - - -
AhriSquare Ahri's Orb of Deception Orb of Deception Unable to test - - -
AhriSquare Ahri's Charm Charm Unable to test - - -
AkaliSquare Akali's Twilight Shroud Twilight Shroud Flashable Location - -
AmumuSquare Amumu's Bandage Toss Bandage Toss Flashable Location - -
AmumuSquare Amumu's Curse of the Sad Mummy Curse of the Sad Mummy Flashable - - -
AniviaSquare Anivia's Flash Frost Flash Frost Unflashable - - -
AnnieSquare Annie's Incinerate Incinerate Unflashable - - -
AnnieSquare Annie's Summon- Tibbers Summon Tibbers Flashable Location Out-Range -
AsheSquare Ashe's Volley Volley Flashable Flash - -
AsheSquare Ashe's Enchanted Crystal Arrow Enchanted Crystal Arrow Unflashable - - -
Aurelion SolSquare Aurelion Sol's Starsurge Starsurge Unflashable - - -
Aurelion SolSquare Aurelion Sol's Voice of Light Voice of Light Flashable Direction - -
AzirSquare Azir's Conquering Sands Conquering Sands* Flashable Location Out-Range -
AzirSquare Azir's Emperor's Divide Emperor's Divide Unflashable - - -
BardSquare Bard's Cosmic Binding Cosmic Binding Unable to test - - -
BardSquare Bard's Caretaker's Shrine Caretaker's Shrine Unable to test - - -
BardSquare Bard's Magical Journey Magical Journey Unable to test - - -
BardSquare Bard's Tempered Fate Tempered Fate Unable to test - - -
BlitzcrankSquare Blitzcrank's Rocket Grab Rocket Grab Cannot - - -
BlitzcrankSquare Blitzcrank's Static Field Static Field Unflashable - - -
BrandSquare Brand's Sear Sear Flashable Location - -
BrandSquare Brand's Pillar of Flame Pillar of Flame Flashable Location Out-Range -
BraumSquare Braum's Winter's Bite Winter's Bite Unable to test - - -
BraumSquare Braum's Glacial Fissure Glacial Fissure Unable to test - - -
CaitlynSquare Caitlyn's Piltover Peacemaker Piltover Peacemaker Unable to test - - -
CaitlynSquare Caitlyn's Yordle Snap Trap Yordle Snap Trap Unable to test - - -
CaitlynSquare Caitlyn's 90 Caliber Net 90 Caliber Net Unable to test - - -
CamilleSquare Camille's Hookshot Hookshot Unflashable - - -
CassiopeiaSquare Cassiopeia's Noxious Blast Noxious Blast Flashable Location Out-Range -
CassiopeiaSquare Cassiopeia's Miasma Miasma Flashable Location In-Range -
CassiopeiaSquare Cassiopeia's Petrifying Gaze Petrifying Gaze Flashable Direction - -
Cho'GathSquare Cho'Gath's Rupture Rupture Flashable Location Out-Range -
Cho'GathSquare Cho'Gath's Feral Scream Feral Scream Flashable Location - -
Cho'GathSquare Cho'Gath's Vorpal Spikes Vorpal Spikes Flashable Location - -
CassiopeiaSquare Cassiopeia's Petrifying Gaze Petrifying Gaze Flashable Direction - -
CorkiSquare Corki's Phosphorus Bomb Phosphorus Bomb Unable to test - - -
CorkiSquare Corki's Missile Barrage Missile Barrage Unable to test - - -

 * I wasn't sure whether to include this, but I did just to be thorough.

As of writing this, I do not have Ahri, Bard, Braum, Caitlyn, Corki, Elise, Gnar, Ivern, Jinx, Kennen, Leona, Lissandra, Lulu, Lux, Nami, Oriana, Quinn, Rakan, Rumble, Taliyah, Twitch, Vel'Koz, Vi, Xayah, Xerath, or Zyra. And I don't feel like testing any more at the moment, but I will come back and add more in the future and, if people can test some of the ones I don't have and put them in the comments, that would be nice.

And the unflashability of an ability means its hitbox and damage, not its visuals. Blitzcrank's R is visually flashable, but its hitbox does not flash with him.

Also, I am not going to list abilities like Darius Q, Camille W, or Corki E because you can move while casting them/while they are in effect.

The Delayed Distance column is up there just in case a direction targeted, delayed blink or dash only goes part way when flashed. I can't think of any examples, but still, just in case. It may be removed after all abilities are tested and we find that there are none that don't go full distance.