Farmable Character Shards

To limit break a character to J2 (5 limit breaks) from J1 (0 limit breaks), you will need 25 shards, which takes 8 1/3 days (for 3/day quests). To limit break a character to J3 (15 limit breaks) from J1, you will need 150 shards, which takes 50 days. This assumes you are farming the maximum 3 shards a day and do not get any shards from other sources (such as duplicate summons). Roughly monthly there is a double drop event for story hard quests that typically lasts a week.

Story Hard Quests
Some characters have permanent hard quests for their shards in the story, which are typically limited to farming 3 times a day (1 drop per clear). You can find the farmable unit list by going to the story section, then the hard quest list, then clicking on "Select by Unit". You can also bookmark the story hard quests you run regularly from the story hard quest screen.

Story EX Quests
There are also some limited units who have permanent EX quests for farming their shards in the story. These quests are limited to 1 clear a day (1 drop per clear) and you must own the unit in order to farm these. You need to clear all of chapters 1-4 and the first 2 episodes of chapter 5 to unlock these. For some later released EX quests, you must complete all of chapter 5. Roughly monthly there is a double drop event for story EX quests that typically lasts a week.

Babel Chronicles Soul Shard Quests
There are also some units whose quests for character upgrade shards are in Babel Chronicles. To unlock these quests you need to do the normal mode Babel Chronicles quests for the chapter that the unit is featured in (i.e. for Gerald it would be Red Sparks of Iron). These quests can't be bookmarked from the story interface. Roughly monthly there is a double drop event for Babel Chronicles quests that typically lasts a week.

Resetting Farm Quests
There is also a special item called Memory Access which allows you to reset the daily limit on a specific shard quest. These items are given out on rare occasions and are not plentiful. It is recommended you save them for when a double drop event is active. There is also a special EX Memory Access for the story EX quests which is more rare than the regular Memory Access items.

Free Units
There are also many free units with ways to get large quantities of shards for them (typically an infinite farm but not always). These are not listed below but you can check the List of Free units. Note this also includes some that are tied to specific events and are not permanently farmable.

For More Information

 * Shard Farm Priority