In canon, I'm pretty sure that the "Queen" title is just one that the Faerie Queen made up, and is thus her interpretation of Taylor's shard's importance filtered through her personal delusion rather than anything inherent to how the shards view each other. But since that doesn't seem to be the case in Splintered, let's consider what are likely to be the qualities of "monarch" shards. Best guess? Critical parts of an Entity's structure and processes. Most of the core shards will count, but they're always bundled up as the Entity's sense of self and are never seeded. QA is functionally a full Entity's central nervous system; the shard that administers and controls other shards via "software" and orders. Shaper is their homeostatic system; the fleshweaver that restructures and controls other shards via "hardware" and physical reshaping.
Harvest - whether we go by the "Ciara's delusions" route or the "critical shards" route, is certain to be another monarch. QA and Harvest are the alpha and omega of the cycle, its start and end points. The former determines where every shard is seeded at the beginning of the process, the latter collects them all again when they've budded and grown. QA and Shaper are the central nervous system and homeostatic mechanisms; Harvest is the kidney and liver equivalent; recycling and reclamation.
Dinah's shard will be another. As per canon, every single precognition shard is ultimately split off from hers; they're all lesser variants of it. She's effectively the Entity's eyes and ears; the root of their sensory organs. Contessa's shard; the only one that trumps her, isn't an integrated part of the cycle. Eidolon's is a core shard that shouldn't be part of the cycle. Alexandria's stasis-lock shard and Legend's propulsion are important tools, but not critical ones - there are other ways to get around. Leet's, I suppose, could qualify if it's similar the "master shard" all Tinker records of advanced technology are split off from, equivalent to Dinah's.
Miss Militia is an ordinary shard. Foil and Parian are inter-Entity weapons, not monarchs - valuable, but not critical. Tats is a useful Thinker shard but, again, non-critical. Same with Dragon. Broadcast... Broadcast is a maybe. Depending on exactly what its role in the cycle is... it's certainly used as part of the whole Entity to communicate, which is highly valuable but non-critical, but if in a "proper" cycle it would be kept back and used to issue cycle-wide updates and patches, that would make it a monarch. Call it close-to-monarch rank - a noble, as it were.
I can't think of any others off-hand that are of equivalent importance. It basically has to be something, I feel, that the combined Entity as a whole critically needs not just to carry out the cycle but to function at all. Central nervous system, homeostatic balance, senses, waste recycling - if you can't analogise it to covering one of the major organ groups of a human body all on its own, it probably isn't a monarch (though of course the weirdness of Entity biology makes that a general rule of thumb rather than anything rigorous).