Reasons for deleting

1. The redirect page makes it unreasonably difficult for users to locate similarly named articles via the search engine. For example, if the user searches for "New Articles", and is redirected to a disambiguation page for "Articles", it would take much longer to get to the newly added articles on Wikipedia.
2. The redirect might cause confusion. For example, if "Adam B. Smith" was redirected to "Andrew B. Smith", because Andrew was accidentally called Adam in one source, this could cause confusion with the article on Adam Smith, so the redirect should be deleted.
3. The redirect is offensive or abusive, such as redirecting "Joe Bloggs is a Loser" to "Joe Bloggs" (unless "Joe Bloggs is a Loser" is legitimately discussed in the article), or "Joe Bloggs" to "Loser". (Speedy deletion criterion G10 and G3 may apply.) See also § Neutrality of redirects.
4. The redirect constitutes self-promotion or spam. (Speedy deletion criterion G11 may apply.)
5. The redirect makes no sense, such as redirecting "Apple" to "Orange". (Speedy deletion criterion G1 may apply.)
6. It is a cross-namespace redirect out of article space, such as one pointing into the User or Wikipedia namespace. The major exception to this rule are the pseudo-namespace shortcut redirects, which technically are in the main article space. Some long-standing cross-namespace redirects are also kept because of their long-standing history and potential usefulness. "MOS:" redirects, for example, are an exception to this rule. (Note also the existence of namespace aliases such as WP:. Speedy deletion criterion R2 may apply if the target namespace is something other than Category:, Template:, Wikipedia:, Help:, or Portal:.)
7. If the redirect is broken, meaning it redirects to an article that does not exist, it can be immediately deleted under speedy deletion criterion G8. You should check that there is not an alternative place it could be appropriately redirected to first and that it has not become broken through vandalism.
8. If the redirect is a novel or very obscure synonym for an article name that is not mentioned in the target, it is unlikely to be useful. In particular, redirects in a language other than English to a page whose subject is unrelated to that language (or a culture that speaks that language) should generally not be created. (Implausible typos or misnomers are candidates for speedy deletion criterion R3, if recently created.)
9. If the target article needs to be moved to the redirect title, but the redirect has been edited before and has a history of its own, then the title needs to be freed up to make way for the move. If the move is uncontroversial, tag the redirect for G6 speedy deletion, or alternatively (with the suppressredirect user right; available to page movers and admins), perform a round-robin move. If not, take the article to Requested moves.
10. Shortcut

    • WP:RETURNTORED
    If the redirect could plausibly be expanded into an article, and the target article contains virtually no information on the subject.

Reasons for not deleting

1. They have a potentially useful page history, or an edit history that should be kept to comply with the licensing requirements for a merge (see Wikipedia:Merge and delete). On the other hand, if the redirect was created by renaming a page with that name, and the page history just mentions the renaming, and for one of the reasons above you want to delete the page, copy the page history to the Talk page of the article it redirects to. The act of renaming is useful page history, and even more so if there has been discussion on the page name.
2. They would aid accidental linking and make the creation of duplicate articles less likely, whether by redirecting a plural to a singular, by redirecting a frequent misspelling to a correct spelling, by redirecting a misnomer to a correct term, by redirecting to a synonym, etc. In other words, redirects with no incoming links are not candidates for deletion on those grounds because they are of benefit to the browsing user. Some extra vigilance by editors will be required to minimize the occurrence of those frequent misspellings in the article texts because the linkified misspellings will not appear as broken links; consider tagging the redirect with the ((R from misspelling)) template to assist editors in monitoring these misspellings.
3. They aid searches on certain terms. For example, users who might see the "Keystone State" mentioned somewhere but do not know what that refers to will be able to find out at the Pennsylvania (target) article.
4. Deleting redirects runs the risk of breaking incoming or internal links. For example, redirects resulting from page moves should not normally be deleted without good reason. Links that have existed for a significant length of time, including CamelCase links (e.g. WolVes) and old subpage links, should be left alone in case there are any existing links on external pages pointing to them. See also Wikipedia:Link rot § Link rot on non-Wikimedia sites.
5. Someone finds them useful. Hint: If someone says they find a redirect useful, they probably do. You might not find it useful—this is not because the other person is being untruthful, but because you browse Wikipedia in different ways. Evidence of usage can be gauged by using the wikishark or pageviews tool on the redirect to see the number of views it gets.
6. The redirect is to a closely related word form, such as a plural form to a singular form.

Neutrality of redirects

Just as article titles using non-neutral language are permitted in some circumstances, so are such redirects. Because redirects are less visible to readers, more latitude is allowed in their names, therefore perceived lack of neutrality in redirect names is not a sufficient reason for their deletion. In most cases, non-neutral but verifiable redirects should point to neutrally titled articles about the subject of the term. Non-neutral redirects may be tagged with ((R from non-neutral name)).

Non-neutral redirects are commonly created for three reasons:

1. Articles that are created using non-neutral titles are routinely moved to a new neutral title, which leaves behind the old non-neutral title as a working redirect (e.g. Climategate → Climatic Research Unit email controversy).
2. Articles created as POV forks may be deleted and replaced by a redirect pointing towards the article from which the fork originated (e.g. Barack Obama Muslim rumor → deleted and now redirected to Barack Obama religion conspiracy theories).
3. The subject matter of articles may be represented by some sources outside Wikipedia in non-neutral terms. Such terms are generally avoided in Wikipedia article titles, per the words to avoid guidelines and the general neutral point of view policy. For instance the non-neutral expression "Attorneygate" is used to redirect to the neutrally titled Dismissal of U.S. attorneys controversy. The article in question has never used that title, but the redirect was created to provide an alternative means of reaching it because a number of press reports use the term.

The exceptions to this rule would be redirects that are not established terms and are unlikely to be useful, and therefore may be nominated for deletion, perhaps under deletion reason #3. However, if a redirect represents an established term that is used in multiple mainstream reliable sources, it should be kept even if non-neutral, as it will facilitate searches on such terms. Please keep in mind that RfD is not the place to resolve most editorial disputes.