HTTP methods beyond GET, POST, PUT, and DELETE can provide more flexibility in web development, but also introduce complexity. PATCH is used to update parts of a resource without replacing the entire thing, while CONNECT requests are used to pass encrypted traffic through proxies. OPTIONS requests help determine allowed CORS requests by making an OPTIONS call to figure out if the request is acceptable. HEAD requests return only the headers and not the body, which can be useful for caching purposes. TRACE requests echo back the input to the sender, but it's considered a security vulnerability due to potential Cross-Site Tracing (XST) risks.