This varies across genres, and non- paying markets seem to be far more prevalent in literary and poetry circles than speculative fiction. Here's the thing: a number of literary and poetry journals don't pay because they're a labour of love. For example journals run by universities, staffed by students, without the budget to pay writers. Some of these are more prestigious than others, and sometimes it's worth not getting paid for appearing in a highly regarded journal or magazine. However sometimes these markets are simply a guy with a blog, posting stories for fun.
There are several ways to look for paying markets. Many writing magazines (certainly here in the UK) include a section of listings. The drawback to this is that you have to wade through all of them, whether or not they're relevant. This is also the case with books like The Writers' and Artists' Yearbook and Poet's Market, with the addition that these quickly go out of date.
Fortunately there are a number of ways to search online:
General market databases
Duotrope - Duotrope contains a searchable database that lets you look for markets by genre and pay rate, among other variables. It leans more towards poetry and literary markets, and has a subscription fee of $5 a month.
The Submissions Grinder - Although technically still in beta, this is a a perfectly functional site with another searchable database. It tends to skew more towards speculative fiction since that's where it originated and where most of the user base lies, but has recently started populating poetry markets and does have a fairly big catalogue of non-genre markets. It's also free.
Both sites also allow for tracking of submissions.
Paying Publications allows for very basic search filtering, and allows for searching of paying poetry markets based on whether a poem is new/already published, already under submission somewhere else, and whether or not the poet is established.
Poets and Writers has a basic searchable database - which doesn't allow for searching by pay rate (see above regarding literary markets!)
Blogs and list sites:
There are also a number of blogs and listing sites that are a useful resource even if not searchable. These are often genre specific.
Dark Markets is for horror markets and offers a very basic filter by publication type (anthology, podcast, etc).
My Little Corner is a blog by author Sandra Seamans that focuses on crime and mystery markets.
Womagwriter is aimed at women's magazine fiction.
Ralan is for speculative fiction markets.
Poet Alison runs the Creative Writing Opportunities List, although please note you need to be logged in to Yahoo to access this.
Facebook groups
There are also several groups you can join on Facebook where opportunities are posted, both paying and non-paying.
Science Fiction, Fantasy, and Pulp Markets.
Horror.
Crime, Thriller, Mystery Markets.
Poetry, Fiction, Art.
And this one specifically for paying markets.
So, there are plenty of opportunities out there!
If you want to submit to a non-paying market, that's up to you. Sometimes the prestige or opportunity to support a charity is worth the loss of payment. You might not be interested in publishing for money. But please, whatever you do, don't give your work away for free because you don't think anyone will pay, because that's simply not true.
With thanks to Helena Bell and Dan Stout for additional links. Further suggestions welcome in the comments.