For every question, there is a straight forward, simple, and wildly incorrect answer.
To use this as an example....
Did you mean "unisee" as an *example* of what you mean by a "'uni' word"?
Or did you mean that you actually registered the single domain name unisee.com, and you want to know if the term "unisee", and that term alone, is claimed as a trademark somewhere on the planet?
Or did you want to know as a general principle whether "uni" words are, or are not, trademarks?
One of the problems of "straightforward answers" is that they assume a straightforward question.
Let's take a look at the question:
---
"All I wanted to know is if I would be infringing on any trademarks if I used a "uni" name. "
---
All you want to know is if you would infringe a trademark if you used a "uni" name, and you define "a uni name" as a name that begins with "uni", by giving an example of "unisee.com" just so we know what you are talking about.
Is that the question?
Because if that is the question, then the answer is there are plenty of trademarks that have "uni" in them. Which was the point of the answer above.
The United States Patent and Trademark Database returns something on the order of 1,912 records of pending, registered, cancelled and abandoned applications for words with "uni" in them. How many of them sound or look like "unisee" or might be a basis for someone to make a claim of confusing similarity? Is that the question?
Now, that's just the United States. There are lots of countries in the world, and there are lots of potential common law claims. You want that "straightforward answer" when?
Are you planning on using it for a webpage? In English? If you're flexible on that, then you can make a page about the Unisee in Bremen, Germany, which seems like a great place to hang out:
http://www.vc.org/deutschland/bremen/bremen/ansichten/alpha/unisee_2.html
http://ici.cegep-ste-foy.qc.ca/departements/allemand/Etudiants/H96/LieuxPers/LP41.4T.F.html
Questions like this are *not* simple questions, and you should be very cautious about people who seem to have all of the simple answers to the "simple questions".