Rest – Is an entity body allowed for an HTTP DELETE request

If your action is not idempotent, then you MUST use POST. If you don't, you're just asking for trouble down the line. GET, PUT and DELETE methods are required to be idempotent. Imagine what would happen in your application if the client was pre-fetching every possible GET request for your service – if this would cause side effects visible to the client, then something's wrong.

I agree that sending a POST with a query string but without a body seems odd, but I think it can be appropriate in some situations.

Think of the query part of a URL as a command to the resource to limit the scope of the current request. Typically, query strings are used to sort or filter a GET request (like ?page=1&sort=title) but I suppose it makes sense on a POST to also limit the scope (perhaps like ?action=delete&id=5).

Overall:

Both PUT and POST can be used for creating.

You have to ask, "what are you performing the action upon?", to distinguish what you should be using. Let's assume you're designing an API for asking questions. If you want to use POST, then you would do that to a list of questions. If you want to use PUT, then you would do that to a particular question.

Great, both can be used, so which one should I use in my RESTful design:

You do not need to support both PUT and POST.

Which you use is up to you. But just remember to use the right one depending on what object you are referencing in the request.

Some considerations:

• Do you name the URL objects you create explicitly, or let the server decide? If you name them then use PUT. If you let the server decide then use POST.
• PUT is defined to assume idempotency, so if you PUT an object twice, it should have no additional effect. This is a nice property, so I would use PUT when possible. Just make sure that the PUT-idempotency actually is implemented correctly in the server.
• You can update or create a resource with PUT with the same object URL
• With POST you can have 2 requests coming in at the same time making modifications to a URL, and they may update different parts of the object.

An example:

I wrote the following as part of another answer on SO regarding this:

POST:

Used to modify and update a resource

POST /questions/<existing_question> HTTP/1.1
Host: www.example.com/

Note that the following is an error:

POST /questions/<new_question> HTTP/1.1
Host: www.example.com/

If the URL is not yet created, you should not be using POST to create it while specifying the name. This should result in a 'resource not found' error because <new_question> does not exist yet. You should PUT the <new_question> resource on the server first.

You could though do something like this to create a resources using POST:

POST /questions HTTP/1.1
Host: www.example.com/

Note that in this case the resource name is not specified, the new objects URL path would be returned to you.

PUT:

Used to create a resource, or overwrite it. While you specify the resources new URL.

For a new resource:

PUT /questions/<new_question> HTTP/1.1
Host: www.example.com/

To overwrite an existing resource:

PUT /questions/<existing_question> HTTP/1.1
Host: www.example.com/

Additionally, and a bit more concisely, RFC 7231 Section 4.3.4 PUT states (emphasis added),

4.3.4. PUT

The PUT method requests that the state of the target resource be created or replaced with the state defined by the representation enclosed in the request message payload.