Relatively few agents even ask for exclusivity on a partial manuscript. It's more common on fulls, but even then, not all of them expect to be the only person reading it.
As for how to find them, you want to submit your queries to agents who represent books that are much like your own. A good way to start narrowing this down is to identify the genre of your book. For example, I wrote a book that can be described as historical fiction for its setting (ancient Egypt) or as literary fiction for its narrative style. So I looked for agents who represent historical and/or literary fiction.
AgentQuery.com is a great place to search. You can also search on Publisher's Marketplace -- you'll need to arm yourself with book titles or authors, though, and then you can use the "Who Represents" search feature on PM to find out who represents some of your favorite books and authors.
Once you have a nice, long list of people who represent the kind of book you've written, you need to research each one. You'll have your own criteria for your list. I kept agents on my list only if they'd had major sales in the past year, and only if they'd sold a debut novelist to a major publishing house in the past three years. (You can find all this out on AgentQuery, on agents' web sites, and on Publisher's Marketplace.) These criteria were important to me because my personal goals include getting a relatively large advance for my book and signing a contract with a big publishing house. You will need to adjust your criteria for agents based on your own personal goals.
Once you've pared your list down, then research all names again, this time checking Preditors & Editors to see whether there are any complaints against these agents; checking to see whether they charge any fees; and other red flags. Generally, if you've found agents who are successfully selling books within the past year, you've found reputable individuals who are serious-minded and are not defrauding their clients -- but it's always better to be safe than sorry, so double-check.
It took me about two weeks of research to put together a list of fifty agents to whom I submitted my query letter. So far, this has resulted in five requests (two partials, one partial turned full, and one full directly from the query package.) So you'll also need to be prepared to have a high rejection rate! That's just the way it goes -- agents are human, and not all of them will be keen on your idea.
Good luck! Let us know if you have more questions.
