As a hybrid support/DPS, Blackbird can fit a lot of teams, either by replacing an old support in an existing team, or building an entire team around herself.
Chaos Hyper Carry Team
Chaos, 
Hathor, Blackbird, Flex(Cosmos)
One of the main challenges that faced Chaos teams is how he needed to constantly trigger two reactions to maximize his DPS, while not having in instant Cycle in is kit. That forced him to compete for 
Zero, which is the most demanded character in all teams on top half of the Esper Circle, while Zero had better uses elsewhere.
With Blackbird, this issue is entirely resolved, as Blackbird instant Cycle gives great value for a Chaos team, allowing her to double dip. First, she upgrade the Psyche slot, providing better buffs and more off field DPS. And second by freeing Zero to create even stronger team for the second half, with Anima characters and Zero
A perfect upgrade for Chaos existing teams
Blackbird Hyper Carry Team
Blackbird, Hathor, 
Daffodill, 
Haniel/ 
Fadia
Instead of upgrading existing team, Blackbird can build a team from the ground up for her own, with herself as the main on-field DPS. She trigger one reaction with her skill, and refills the Esper Meter by performing her attacks on field to trigger a second reaction. This way, she can keeps both Nova and Stain on enemies a lot of the time.
So she gets the full triple buffs from Haniel, or the enemy HP reduction from Fadia, all while keep Stain up for extra personal DPS. You can also build Daffodill for full 
Cycle Intensity to maximize Nova damage while building Blackbird as typical ADC carry to max skill damage.
You can ditch chaos characters and switch to a mono Stain team as well, but you lose a good part of team mates kits, as all other Psyche characters enhance Nova reaction, not Stain. While Lakshana characters buff Remora.
Lacrimosa Discord Team
Lacrimosa, 
Zankou, Blackbird, 
Sakiri
The only reason why A0 Lacrimosa wasn't placed as the absolute top DPS in the game was how hard to utilize her hardest hitting skill, her burst attack with Discord reaction. That's because Discord also need two constant reactions.
Now with Zankou auto reapplying Scorch, and Blackbird having instant Cycle to apply Nova, Lacrimosa will have so many Discord triggers in a fight, and her damage will peak better than ever. This is true at A0, but at A1 it will be even better, she can spam Nova and Discord non-stop, that she can overwhelm any boss, have huge invulnerability frames, and quickly clear anything in her way.
Zankou DoT team
Zankou, Sakiri, Daffodill, Blackbird
It's almost similar to Lacrimosa team, but with Zankou as the main DPS instead of Lacrimosa. At that point, a lot of characters competes for field time, so Daffodill quick swap style fits this team more, and Blackbird stays as a support.
This team can focus on break a lot easier, due to constant Discord triggers and Break bonuses from Daffodill's kit. But it wouldn't be an upgrade in place of 
Linko in that fourth slot, as Zankou's own DPS is more important than Discord in such a team. So think of it as a sidegrade, or another flavor.
You may also use Linko in this team in place of Sakiri for another variation.
Mono Nova Team
Lacrimosa, Blackbird, 
Hotori, Fadia/ Haniel
The classic A1 Lacrimosa team with constant Nova reaction with Hotori to replicate the skill of Lacrimosa to get an additional free Nova every 30 seconds.
Now with Blackbird, Hotori can replicate both of Blackbird and Lacrimosa skills, resulting in two extra reactions every 30 seconds, instead of one. Also Blackbird have higher Cycle Intensity potential , thanks to 75 more from her signature weapon. So the reaction itself will get higher damage, and you can trigger up to total 7 or 8 reactions in a rotation!
Blackbird
Blackbird is S-Class character, with Psyche element who uses a Gas Arc. She's the first limited banner character in v1.4, and 9th banner character in total.
