Build Overview

Muriel is underrated as heck, and her sustain with this build is simply ungodly. Her level 1 does absurd damage, and she scales very well into the later game, making anyone you're supporting virtually unkillable while ensuring that the damage just keeps coming.

Crest Options

• Tranquility (Recommended): Thanks to the short CD and powerful healing throughput, Tranquility remains an amazing source of healing. Use this as often as possible with as many allies nearby as possible. In a pinch, it can be handy for an emergency heal, but don't save it just for that; you'll usually get more sustain value using it off CD. Burst heal go BRRRR.
• Reclamation (Teamfights/Mass Cleanse): A mass cleanse button, even on such a long CD, is very impactful in big team fights. Particularly good if the enemy comp has a ton of heavy CC or DoTs, but it takes some more attention to detail to use it effectively, and isn't usually optimal on Muriel.
• Silentium (Disruption/CC Counter): Silentium is very much a counter-build flex item. Pick against characters with long CC ults and abilities (Gideon, Riktor, Narbash, etc.).

First Item

Lunaria should almost always be first. Muriel's early-game damage and poke potential are extremely high, meaning she gets extremely good use out of Lunaria early on. This also doubles down on the already impressive sustain from her E (which has replaced her Q as the best early-game ability to max).

It's also worth noting that you should probably go for Codex as a first buy, as the extra magic damage helps capitalize on Muriel's ridiculous damage at Level 1 (hence why one should always buy her Q first, regardless of which ability you choose to prioritize maxing).

Second Item

Everbloom is just generally the best second item on Muriel support; her frequent shields paired with the damage mitigation make for an extremely powerful sustain combo.

Alternatives

• Tainted Totem is a counter-build flex here (in which case it replaces Windcaller later on, pushing Everbloom, Astral Catalyst, and Crescelia into the 3rd, 4th, and 5th slots) when facing a Narbash or Phase in-lane. Typically you won't need anti-heal until later in the game except against those heroes, however.
• Crystal Tear is a flex option when alongside a Twinblast, Grim, or (occasionally) Wraith, replacing Tainted Totem in your build. If you need Tainted Totem, you probably shouldn't build Crystal Tear.

Third & Fourth Items

Note: these will also possibly be your fourth and fifth items if you slotted Tainted Totem in early.
In order:

1. Astral Catalyst is a large boost to magic power, boosting both damage and sustain. More importantly, however, it allows you to make more frequent use of your ult to engage (so long as you're getting takedowns off of it), doubling down on her map-wide presence.
2. Crescelia plays further into Muriel's identity as an aggressive sustainer, allowing more damage baseline to stack Lunaria as well as allowing for much more frequent ability spam.

Final Items

At this point, you have a few different items to round out the game with. I typically pick from a couple of different items here:

1. Tainted Totem is probably what you'll end up building in the 5th slot in most games (assuming you didn't flex it to 2nd), since anti-heal is such a must-have in the current meta. One could think about going Tainted Scepter or something else entirely instead (especially if most of the rest of your team has anti-heal already), but Totem is just consistently reliable regardless of the state of play.
2. Vanguardian is your go-to final item if the enemy team frequently tries to single you out, giving both you and everyone around you (i.e. your protectors) a bit more tankiness to keep both you and them alive when jumped.
3. Oblivion Crown is your go-to final item if you're not being targeted, allowing you to double down in equal part on insane levels of shield sustain and free damage.

Alternatives:

• Windcaller works as a fantastic 5th (or, less commonly, 6th) item boost for hyper-aggressive play. However, this also requires your carry to be able to match that level of aggression. If you're playing more defensively or having trouble staying ahead, this doesn't work as well.