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The trivial representation maps all group elements to the identity transformation. This representation exists for any group.
Spin-$0$ particles/fields transform under the trivial representation of $SU(2)$. That is, their spin value does no depend on orientation in space.
The action (and often the Lagrangian) transforms under the trivial representation of space-time and gauge symmetry groups. That is, the action does not depend on these symmetry transformations.
Example
The diagram below shows the defining representation of $SU(2)$ in its upper branch and the trivial (1-dimensional) representation in its lower branch.
For a more detailed explanation of this diagram see Fun with Symmetry.