Excalibur
As the Prime Shard of Honor, Excalibur follows its own rules and has its own expectations. People who might not carry a Shard, like Skinner, can qualify for the blade.
What does it take? The wielder of Excalibur must be willing to step into the archetypical role of King: they must be willing to fight, lead, and rule over the land which they claim.
Those things are defined very loosely - William was able to declare an asteroid as his “land” and teleport there on the basis of the principle that “the land and the king are one”, but it also bound him to protect it until the people there were satisfied.
Unfortunately, none of the core Menagerie members qualify! They’re willing to fight - but maybe not to kill, something that many of the Sword’s wielders have no problem with. They can lead to a greater or lesser extent. But I feel like none of them will rule - taking responsibility and authority over large groups of people, issuing edicts and orders, standing as the king.
Charlotte in particular has rejected this kind of leadership explicitly, after reluctantly accepting it once.
Alycia, likewise, explicitly walked away from this sort of role. She’s relegated herself to leading a strike team, taking direction from others whose moral stances she can take or leave as the situation dictates.
The Marvelous Menagerie
Nautilus doesn’t have the requisite virtue or moral center but seems content to make decisions for large numbers of people, and his son Stingray has rejected that view. Ninjess, likewise, is a humble ninja.
A10 of all people could conceivably take up the sword, but is unlikely to be offered it at this time.
Menagerie In Action
Nobody on the black ops team makes the cut, unsurprisingly.
Menagerie at Midnight
Of anyone, Equity might qualify for the sword. She’s attracted the notice of one god, and the cosmic principle behind Excalibur might likewise find her interesting.
Newmenagerie
None of the Newmen are willing to rule, although all of them are plenty ready to fight and have demonstrated leadership ability. Of all of them, Aria comes closest to qualifying, as her goals for Safe Harbor and her willingness to do whatever it takes to keep her people safe could put her on the path of Queen.
Menagerie Megaverse
William is the canonical wielder of Excalibur, and Jaycee has been shown to qualify in a future timeline. But right now, Jaycee doesn’t want the sword for complicated personal reasons, and the sword will not offer itself to a disinterested candidate - unlike gods like Palamedes or the Concordance Shard, which glommed itself onto Adam without even asking.
Jordan, Keri, and Space Bug all lack the willingness or ability to lead and rule that the sword would demand.
Others
Unlike the Concordance, who seems able to issue Shards to planets or individual people, the Sword chooses its own wielders in their moments of need. It will withdraw that honor as well - the Dread Moor Velasco once bore the blade, and now carries all the trappings of an archetypical King, but is no longer worthy of bearing it for reasons as yet unknown.