"It seems [like] a random occurrence"

Solution 1:

All the sentences you wrote can be used, and would mean to give the impression or sensation of being something or having a particular quality.

Dawn seemed annoyed.
There seems to be plenty to eat.
It seemed that he was determined to oppose her.

Looking on the Corpus of Contemporary English for some phrases containing it seems, I get the following result:

Phrase         |  Frequency per million
               |
               |   Spoken | Magazine | Newspaper | Academic
It seems a     |    1.11  |  0.93    |  1.09     |  0.66
It seems like  |   17.33  |  3.94    |  6.68     |  0.76
It seems that  |    9.40  |  6.30    |  5.35     | 12.01
It seems to be |    6.90  |  1.90    |  2.32     |  1.87