What makes a compelling magic system in fantasy writing?
A compelling magic system in fantasy writing should be well thought out, consistent, and integrated into the world-building of the story. Here are some key elements that can make a magic system more compelling:

  1. Rules and limitations: A magic system that has clear rules and limitations is more believable and easier for readers to understand. For example, magic might require certain incantations, gestures, or materials, and it might have limitations on who can use it, how much can be used, and what it can do.

  2. Unique and memorable: A magic system that is unique and memorable can help the story stand out in a crowded field. This could be achieved through creating new types of magic, combining existing elements in new ways, or introducing twists on traditional magical concepts.

  3. Internal logic: A magic system should have internal logic that makes sense within the context of the story. For example, if magic is based on manipulating energy, then it should be consistent and follow the laws of physics within the story's world.

  4. Tied to the world-building: A compelling magic system should be integrated into the world-building of the story. It should have a history and a cultural significance, and it should impact the societies and characters within the story.

  5. Characters and conflict: Magic should create opportunities for interesting characters and conflict. It could be used as a source of power or weakness, create ethical dilemmas, or be a point of contention between characters.

Overall, a compelling magic system should be imaginative, well-crafted, and enhance the story rather than distract from it.
Could you give me a positive and a negative example for each of the above-mentioned key elements ?
Sure. However, it is important to know that the feeling towards the magic systems in works can vary a lot between individuals, and my answers might be inaccurate for you.

1. Rules and limitations
Positive: Fullmetal Alchemist. It provides taboos and principles of Alchemy, as well as stories related to following the rules or going beyond and braving the consequences.
Negative: Iron Kingdoms: Full Metal Fantasy. The magic can be used in combats, but not in other scenarios since there are no rules to direct magic users and stop them from spamming.

2. Unique and memorable
Positive: Simon Snow. It has a creative magic system where common sayings, song lyrics, and pop culture references are used to cast spells, and more well-known ones can improve the effect. 
Negative: The Marvel Cinematic Universe. It can be exciting to give each hero a unique suite of powers. However, this sometimes makes the story themed like a mess of ideas stuck together in a hurry, making each of the powers less memorable.

3. Internal logic
Positive: The Inheritance Cycle. The magic system works more like a programming language, and the conservation of energy between casting a spell and taking the same action without magic is also talked about a lot.
Negative: Vampire: the Masquerade. The Thaumaturgy rules lack a consistent logic to separate different sets of paths and rituals, making it difficult to talk about what makes a path special with logical reasons.

4. Tied to the world-building
Positive: Alex Stern. The society is heavily influenced by the magic system, with eight different societies each specilize in different magic, the depravity caused by magic, and dark rituals described in detail.
Negative: Fate/Stay Night. It actually has explained how the Holy Grail War is kept secret reasonably. However, as the effect and collateral damage of fights become much greater in the animes, the fact that nobody in a modern society has noticed this also becomes a little weirder.

5. Characters and conflict
Positive: Godless. Magical ability is not only a benefit, but also a sign of infection from something corrupted by a dead god that can cause a horrible death, putting the character into a crisis.
Negative: Middlegame. The characters tried solving major problems with magical math formulas, but the process is not well explained with too many terminologies not introduced in detail. This makes the turning point less satisfactory and exciting.
If magic has logic to it doesn't that ruin the 'feeling' of the unknown and vast possibility that lies within it. I like the idea of my characters discovering new and unknown aspects of magic while balancing the usefulness of what they learn but how would I make my magic system logical where no one else (or very few) have discovered these things yet my characters have?