A so-called Trash Sense is a Sense that overall player base (generally the Beta Players) agree is not worth obtaining, due to one reason or another. The perception of which Senses are Trash can change over time. Beta players put together a list of non-worthwhile Senses before the official launch occurred, so many new players used that list to determine which Senses to avoid. On the other hand, Yun went to that list and used it to decide which Senses to take.
A Sense can be considered a Trash Sense for any number of reasons. The following are a few examples of which Senses were considered so, and why:
Bow was considered a Trash Sense because of its history in the Beta, nearly causing everyone to lose an event due to its heavy dependence on consumables and low accuracy rate (because most players sought after increasing ATK instead of DEX) compared to magic
Cooking was considered a Trash Sense because it required a lot of effort for only minor, temporary gains and, as such, was less profitable compared to other crafting Senses
Taming was considered a Trash Sense because of its abysmally low success rate and its difficulty in leveling
Hawk Eyes was considered a Trash Sense because its main functions were either minor or could be accomplished with better Senses, and its other functions never had a chance to be noticed
Enchant was considered a Trash Sense because it had such a low range that the backline players who could use it were usually too far away from the frontline players who could benefit from it
However, just because a Sense is considered "Trash" doesn't mean it is useless or can't be effectively used. The downsides of each can be overcome with in-game information, experience, knowledge, or by a patch from the developers.
For example, the Bow Sense becomes more efficient when paired with a crafting build because of its DEX dependency and the ability to craft and improve arrows. The Alchemy Sense has unique functions that other crafting Senses don't have. Cooking was patched with the Summer Event to become necessary by making hunger a mechanic. With the Summer Event and later patches, Taming was given another chance by Young Beasts being added and given away, making it easier to level up. Hawk Eyes is uniquely effective when combined with magic Senses, including Enchant.