Katya - Rhapsody

If you don't have Katya - Dawnwing yet but want to try her out, she is available as a Trial Operative in Story Mission 15B-9

Rhaapsody has a staggeringly high rate of fire that gives her strong damage potential, her jetpack can give her good mobility, and she can be capable against both single-targets and groups.
Her mobility in actual combat can be quite low though as her firing arcs are limited by the amplifier created by her own standard skill, and the amplifier itself means that Rhapsody can struggle to deal full damage against enemies that are very mobile or in melee range, and being tied to her amplifier means her mobility isn't good either.
Overall Rhapsody is a very capable operative with a few small caveats.

Overview

Role: Main DPS
Deals Frost Damage
Deals both ballistic and skill damage
Rhapsody deals mostly ballistic damage
Her skill damage is also buffed by ballistic buffs
Mobility is:
In combat: Terrible
Otherwise: Very good

How to use Katya - Rhapsody

Hold down the trigger.
If your standard skill isn't active, use it.
If her ult is ready, use it

How to build Katya - Rhapsody

Weapon: Cobalt Cyclone / Infinite Chronos
Weapon Parts: Max ATK
Logistics Squad:
+10.0% ATK
+106 Alignment Index
+8.5% Frost Damage
Best supports:
Tess - Phantasia
Vidya - Celestial Swan

Standard Attack
Rhapsody's minigun has a RoF of 1,800 rounds per minute, equivalent to 30 rounds per second.

Rhapsody's minigun also has a short spin-up time - a small delay after you hold the trigger but before Katya starts shooting.

Passive Effects

Rhapsody can't use cover.

Rhapsody has what is basically a jetpack that she automatically uses instead of sprinting to move faster.
When full, Rhapsody has enough fuel for 6 seconds of jetpack usage.
The jetpack fuel gauge is in the lower-left, above the ammo counter.
Rhapsody gains 1.5s of jetpack fuel after dodging.
If her standard skill isn't up, using her jetpack also recovers S-Energy for her.

Standard Skill

Rhapsody's standard skill creates a "Dimensional Amplifier" (i.e. big screen with guns on it) in front of herself, which drains her S-Energy and prevents S-Energy regen.
It uses 0.7 S-Energy every 0.1s, so 7 S-Energy per second, or a 7.1 second duration.
Rhapsody's shots that go through the amplifier cause the amplifiers guns to also shoot at the target, up to every 0.2s.
Since Rhapsody has a RoF or 30 rounds per second, the cannons 5 shot per second RoF means they fire 6x slower than Rhapsody herself does.
Their low RoF means that most of Rhapsody's damage comes from her own minigun, not her standard skill.
The amplifiers cannons are buffed by hipfire ballistic damage buffs, but can't crit.

After shooting through the amplifier enough times, the amplifier improves, and Rhapsody shooting through the amplifier also grants Rhapsody a +30% ballistic dmg buff for 1s.
The amplifier can go up to phase 3.
With its first Neuronic buff, the amplifier at phase 3 also grants Rhapsody 7 bullets back whenever the cannons fire, and gives her 50 bullets back when the maxed amplifier ends.
Rhapsody shoots 30 shots per second, and the amplifier triggers every 0.2s, or 5 times per second, so Rhapsody gains 5*7=35 shots per second.
Rhapsody loses bullets until the amplifier is maxed, but slowly gains them after that. Overall this means Rhapsody can sustain herself well.
Provided she only shoots while her standard skill is active, this does give Rhapsody unlimited firing time.
Even with somewhat non-optimal play, Rhapsody will basically always kill the enemy before she runs out of ammo.

If the barrier is already active, using Katya's standard skill again just moves the barrier in front of her, which doesn't interrupt her or stop her shooting.
Since her standard skill has an S-Energy cost of 0, this doesn't alter how long she can sustain her amplifier for.
This allows Rhapsody to repositions, but significant repositioning might require using the skill several times.

When Katya runs out of S-Energy, holds her standard skill, or dodges through the barrier, it ends, which refunds Katya's S-Energy.
This allows Rhapsody very high uptime on her standard skill.

The amplifiers cannons are not auxiliary units, and hence aren't buffed by Auxiliary Unit buffs.

Ultimate Skill

Rhapsody floats in mid-air and has her cannons bombard the area in front of her for 7s.
This fires very quickly, and each hit can damage up to 10 enemies.
Her four cannons fire every 0.2s, and the 2 cannons fire every 0.35s. Each weapon type deals one instance of damage to an enemy per activation.
I.e. The four cannons firing deals one instance of damage each to up to 10 enemies, and the two cannons deal one instance of damage each to up to 10 enemies.

The amplifiers cannons are buffed by hipfire ballistic damage buffs, but can't crit.

Defeating an enemy using Rhapsody's ult increases the ults duration by 2s, up to the 7s max.
There is no ICD or limit on this, so with enough enemies to defeat Rhapsody can use her ult literally indefinitely.

Rhapsody's ult has an extremely low cooldown of 5s, and a very low U-Energy cost of 12, making it extremely spammable.
With her second Neuronic, Rhapsody reduces the U-Energy cost of her ult by 1 for each enemy defeated, and the cost can go to 0.

With its first Neuronic, Rhapsody's ult is free every 10 mins. I.e. her first ult in a map is free.
Given how low its U-Energy cost is, in practice this isn't a massive buff to her.

Support Skill

Rhapsody deals frost damage and knocks targets up into the air and deals more damage as they fall (38 damage instances total)
This can't crit, and is affected by ballistic damage buffs.

The skill has a tiny S-Energy cost of 5, but a very long cooldown of 15s, although with her Neuronic defeating a target with the skill reduces its cooldown by 5s.

Like all main-dps operatives, Rhapsody is not a good support and you really don't want to use her as one.

Weapon

Cobalt Cyclone
Her 5* weapon, and her best weapon.
Recovers 0.5 S-Energy every 6 shots.
Since Rhapsody has a RoF of 30 shots/second, this gives 2.5 S-Energy per second, increasing her standard skills duration from 7.1s to 11.1s, which is great.
This gives ballistic damage, which is good.
Like all modern 5* weapons, this also increases the damage the enemy takes, which is very strong.
Infinite Chronos
Her 4* event weapon.
Recovers 0.3 S-Energy every 6 shots.
Since Rhapsody has a RoF of 30 shots/second, this gives 1.5 S-Energy per second, increasing her standard skills duration from 7.1s to 9s, which is nice.
It also gives a nice ballistic damage buff after a skill hit, which is nice.

For weapon Parts just go with max ATK.

Logistics

Use Nebula squad - it was designed specifically for her and is her best choice.

For stats Rhapsody wants 10.0% ATK and +106 Alignment Index.
For the Special talent, Rhapsody wants 8.5% Frost Damage.

ATK for more damage.
Alignment Index also gives her more damage.

Crit DMG Amplifier only adds onto the weapons base crit dmg, which for Rhapsody is only 10%.
This means that even with 3x maxed 10.6% Crit DMG Amplifier rolls, the damage of Rhapsody's crits will go from 110% to 113.18%, which is terrible for three maxed-rolled stats.
Similarly, because her crit damage is always so low, any buffs to her crit rate are also extremely weak.
Crit Rate and Crit DMG Amplifier will increase her damage, but both are extremely weak cope stats and are, quite frankly, mostly useless.
Rhapsody's skills can't crit either, meaning they don't benefit from any crit buffs, making crit buffs even less useful.

Since she regains her S-Energy when her standard skill ends, S-Energy regen isn't overly useful for her.

U-Energy regen doesn't help Rhapsody due to just how low her ults U-Energy cost is, but U-Energy regen can help get more U-Energy for her supports.
Since Rhapsody's main supports (Phantasia and Celestial Swan) also don't care for U-Energy, so giving them more U-Energy doesn't really help them either.
U-Energy regen is something that Rhapsody does benefit from, but it's borderline useless.

As such, any basic stat other than ATK or Alignment Index is basically useless for Rhapsody.

Manifestation

M1 causes enemies hit by her standard skills cannons to take 18% increased ballistic damage for 1s.
Like all increased damage taken mods, this is very strong.
M2 increases the full squads Frost DMG by 20% when her standard skill is at phase 3.
This is an ok buff for Rhapsody herself, but it also buff's the damage from Tess - Phantasia, which is nice.
M3 causes Katya's shots that have gone through her amplifier to reduce the targets DEF by 1% for 1s, up to 15%.
A 15% DEF reduction is quite a nice buff.
M4 gives the usual +1 level to her standard skill, as well as increasing its damage by 10%.
A nice increase, but it doesn't change how she plays, only increasing her damage.
M5 increases Rhapsody's ATK by 10%, and a further 10% if there is another Frost operative on the squad.

As usual, if you want efficiency (good damage without massive investment), then M3 is a good stopping point.
Given how strong her M1 is, and her M2 as well if you have Phantasia, M2 is also a decent stopping point. M1 isn't a bad stopping point either if you don't have Phantasia.

Good Supports for Katya - Rhapsody

Most of Rhapsody's damage comes from her own ballistic damage, with a little skill damage, but she does deal skill damage quite frequently.

Whilst Rhapsody does deal skill damage, all of her damage is buffed by ballistic damage buffs.
Her skills damage can't crit, although Rhapsody's crit damage is so low that that doesn't really change the fact that Rhapsody doesn't care much for crit buffs regardless.

Good supports for Rhapsody:

Tess - Phantasia
Phantasia adds Frost damage to each shot, which Rhapsody does a lot of, allowing Tess to add a vast amount of damage to Rhapsody.
Phantasia is also on-element to Rhapsody, and can thus be used with Celestial Swan, and also proc Rhapsody's M5 buff.
Vidya - Celestial Swan (and other resistance shredders)
Celestial Swan and other resistance shredders (Acacia - Kaguya, etc.) that reduce the enemies resistance help enemies take more damage.
If using Celestial Swan, make sure the other support isn't off-element to the main DPS.
Kaguya also deals Frost damage, and thus Celestial Swan + Kaguya does work, although sinc ethey're both resistance debuffers that's not an ideal team.
Fritia - Little Sunshine
The sheer rate of fire that Rhapsody puts out means that Little Sunshine can add a lot of damage to Rhapsody, and even though she's a 4* Fritia is very strong for Rhapsody.
Fritia is off-element to Rhapsody.
Fenny - Starshine
Starshine gives Rhapsody a lot of bullets, and Rhapsody benefits a lot from Starshine's ballistic buffs.
Enya - Exuvia
Rhapsody doesn't really benefit from Exuvia's skill damage buffs, but Rhapsody's rate of fire and constant skill damage hits allow her to maximize Exuvia's damage
Exuvia is off-element to Rhapsody.
Yao - Nightglow
Rhapsody benefits from Yao's ATK and ballistic buffs, although Yao's crit buffs aren't overly useful for Rhapsody, and Yao is off-element to Rhapsody..
Eatchel - The Cub
Rhapsody's a relatively static operative, which makes her vulnerable to taking damage, so Eatchel's healing can be useful, and Eatchel's ATK buffs are good.
Eatchel doesn't deal damage, but is technically a Kinetic-damage operative, and hence doesn't proc Katya's M5 buff.
Yao - Quiet Quitter
If you need a healer and don't have Eathcel yet, then you'll want to use Yao.
Since Yao is on-element she can proc Rhapsody's M5, although if you've gotten Rhapsody to M5 you'll hopefully also have Phantasia as well as not need a healer any more.

Rhapsody's best supports are Tess - Phantasia and Vidya - Celestial Swan.
Rhapsody + Kaguya is also a good combo.