What are the examples of trivial and non-trivial properties of language recognized by a Turing machine?
In the context of Rice's theorem, a trivial property has a very specific meaning.

Here, we define a property to be a set of Turing-recognizable languages. A property is non-trivial unless it contains no languages, or contains all Turing-recognizable languages.

We can clearly recognize these with a Turing Machine, since we can just make a machine that always says "YES" for the first case, or make a machine that always says "NO" for the second.

So both the properties you have identified are non-trivial properties, and are thus undecidable by Rice's theorem.