It is possible to attach files to Wiki pages ... See : http://www.redmine.org/projects/redmine/wiki/RedmineProjects for example
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Sorry, I forgot to mention I meant via the REST API.
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Had the same problem here, the following patch make it working (I am not a ruby specialist, just took the snippet from the issue controller; normal upload and rest upload is then working):

<pre><code class="diff">
Index: wiki_controller.rb
===================================================================
--- wiki_controller.rb (revision 14728)
+++ wiki_controller.rb (working copy)
@@ -152,7 +152,7 @@
@content.author = User.current

if @page.save_with_content(@content)
- attachments = Attachment.attach_files(@page, params[:attachments])
+ attachments = Attachment.attach_files(@page, params[:attachments] || (params[:wiki_page] && params[:wiki_page][:uploads]))
render_attachment_warning_if_needed(@page)
call_hook(:controller_wiki_edit_after_save, { :params => params, :page => @page})
</code></pre>

Added this patch to the patch-queue #21066
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I added a test-case for Sven S.'s patch.
This patch workes fine under the latest trunk r15317.
I would be happy if I could see the feature in Redmine 3.3.0.
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Committed, thanks. I've removed the test with the 'attachment' param as it's not supposed to be used with the API. Attachments should be added through the API with wiki_page[uploads].
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Michael Medin wrote:
> Currently (2.3.2) it is not possible (that I know of) to attach files to wiki pages.
> AS a workaround i have attached all files to an arbitrary ticket instead which works but is pretty awkward.
>
> It would be nice if it was possible to attach the file directly to the wikipage instead.
>
> // Michael Medin

I'm realy not clear about this feature.@Michael, Lang Can you explain more about it. How can i replay it ? I'm using v.3.2.6
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Vu Anh wrote:
> Michael Medin wrote:
> > Currently (2.3.2) it is not possible (that I know of) to attach files to wiki pages.
> > AS a workaround i have attached all files to an arbitrary ticket instead which works but is pretty awkward.
> >
> > It would be nice if it was possible to attach the file directly to the wikipage instead.
> >
> > // Michael Medin
>
>
> I'm realy not clear about this feature.@Michael, Lang Can you explain more about it. How can i replay it ? I'm using v.3.2.6

This feature is included in Redmine 3.3.
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relates,Closed,21066,Enable adding attachments to wiki entries using the Rest-API
duplicates,Closed,13843,Redmine API: can wiki pages contain uploads