diff --git a/lib/rules/component-name-in-template-casing.js b/lib/rules/component-name-in-template-casing.js
index 629f51209..d330f60da 100644
--- a/lib/rules/component-name-in-template-casing.js
+++ b/lib/rules/component-name-in-template-casing.js
@@ -122,9 +122,12 @@ module.exports = {
       }
 
       if (
-        (!utils.isHtmlElementNode(node) && !utils.isSvgElementNode(node)) ||
+        (!utils.isHtmlElementNode(node) &&
+          !utils.isSvgElementNode(node) &&
+          !utils.isMathElementNode(node)) ||
         utils.isHtmlWellKnownElementName(node.rawName) ||
         utils.isSvgWellKnownElementName(node.rawName) ||
+        utils.isMathWellKnownElementName(node.rawName) ||
         utils.isVueBuiltInElementName(node.rawName)
       ) {
         return false
diff --git a/lib/rules/html-self-closing.js b/lib/rules/html-self-closing.js
index 531ddfc2b..5b98d8cf7 100644
--- a/lib/rules/html-self-closing.js
+++ b/lib/rules/html-self-closing.js
@@ -63,7 +63,7 @@ function getElementType(node) {
   if (utils.isSvgElementNode(node)) {
     return 'SVG'
   }
-  if (utils.isMathMLElementNode(node)) {
+  if (utils.isMathElementNode(node)) {
     return 'MATH'
   }
   return 'UNKNOWN'
diff --git a/lib/rules/no-deprecated-html-element-is.js b/lib/rules/no-deprecated-html-element-is.js
index 7a0fd6225..5253ef2b7 100644
--- a/lib/rules/no-deprecated-html-element-is.js
+++ b/lib/rules/no-deprecated-html-element-is.js
@@ -27,7 +27,8 @@ module.exports = {
     function isValidElement(node) {
       return (
         !utils.isHtmlWellKnownElementName(node.rawName) &&
-        !utils.isSvgWellKnownElementName(node.rawName)
+        !utils.isSvgWellKnownElementName(node.rawName) &&
+        !utils.isMathWellKnownElementName(node.rawName)
       )
     }
     return utils.defineTemplateBodyVisitor(context, {
diff --git a/lib/rules/no-undef-components.js b/lib/rules/no-undef-components.js
index 9eb74ad05..4371add7c 100644
--- a/lib/rules/no-undef-components.js
+++ b/lib/rules/no-undef-components.js
@@ -149,6 +149,7 @@ module.exports = {
       if (
         utils.isHtmlWellKnownElementName(rawName) ||
         utils.isSvgWellKnownElementName(rawName) ||
+        utils.isMathWellKnownElementName(rawName) ||
         utils.isBuiltInComponentName(kebabCaseName)
       ) {
         return false
@@ -177,7 +178,11 @@ module.exports = {
     /** @type {TemplateListener} */
     const templateBodyVisitor = {
       VElement(node) {
-        if (!utils.isHtmlElementNode(node) && !utils.isSvgElementNode(node)) {
+        if (
+          !utils.isHtmlElementNode(node) &&
+          !utils.isSvgElementNode(node) &&
+          !utils.isMathElementNode(node)
+        ) {
           return
         }
         verifyName(node.rawName, node.startTag)
diff --git a/lib/rules/no-unused-components.js b/lib/rules/no-unused-components.js
index 75b08fa14..9ba889ccc 100644
--- a/lib/rules/no-unused-components.js
+++ b/lib/rules/no-unused-components.js
@@ -52,9 +52,12 @@ module.exports = {
         /** @param {VElement} node */
         VElement(node) {
           if (
-            (!utils.isHtmlElementNode(node) && !utils.isSvgElementNode(node)) ||
+            (!utils.isHtmlElementNode(node) &&
+              !utils.isSvgElementNode(node) &&
+              !utils.isMathElementNode(node)) ||
             utils.isHtmlWellKnownElementName(node.rawName) ||
-            utils.isSvgWellKnownElementName(node.rawName)
+            utils.isSvgWellKnownElementName(node.rawName) ||
+            utils.isMathWellKnownElementName(node.rawName)
           ) {
             return
           }
diff --git a/lib/rules/require-typed-ref.js b/lib/rules/require-typed-ref.js
index 60a82432a..c05e6c829 100644
--- a/lib/rules/require-typed-ref.js
+++ b/lib/rules/require-typed-ref.js
@@ -44,13 +44,23 @@ module.exports = {
       return {}
     }
 
-    const scriptSetup = utils.getScriptSetupElement(context)
-    if (
-      scriptSetup &&
-      !utils.hasAttribute(scriptSetup, 'lang', 'ts') &&
-      !utils.hasAttribute(scriptSetup, 'lang', 'typescript')
-    ) {
-      return {}
+    if (utils.isVueFile(filename)) {
+      const sourceCode = context.getSourceCode()
+      const documentFragment =
+        sourceCode.parserServices.getDocumentFragment &&
+        sourceCode.parserServices.getDocumentFragment()
+      if (!documentFragment) {
+        return {}
+      }
+      const scripts = documentFragment.children.filter(
+        /** @returns {element is VElement} */
+        (element) => utils.isVElement(element) && element.name === 'script'
+      )
+      if (
+        scripts.every((script) => !utils.hasAttribute(script, 'lang', 'ts'))
+      ) {
+        return {}
+      }
     }
 
     const defines = iterateDefineRefs(
diff --git a/lib/rules/script-setup-uses-vars.js b/lib/rules/script-setup-uses-vars.js
index 703d0cb9a..a1d873a6b 100644
--- a/lib/rules/script-setup-uses-vars.js
+++ b/lib/rules/script-setup-uses-vars.js
@@ -94,10 +94,13 @@ module.exports = {
         },
         VElement(node) {
           if (
-            (!utils.isHtmlElementNode(node) && !utils.isSvgElementNode(node)) ||
+            (!utils.isHtmlElementNode(node) &&
+              !utils.isSvgElementNode(node) &&
+              !utils.isMathElementNode(node)) ||
             (node.rawName === node.name &&
               (utils.isHtmlWellKnownElementName(node.rawName) ||
-                utils.isSvgWellKnownElementName(node.rawName))) ||
+                utils.isSvgWellKnownElementName(node.rawName) ||
+                utils.isMathWellKnownElementName(node.rawName))) ||
             utils.isBuiltInComponentName(node.rawName)
           ) {
             return
diff --git a/lib/utils/index.js b/lib/utils/index.js
index f16bcef2c..9671c00d4 100644
--- a/lib/utils/index.js
+++ b/lib/utils/index.js
@@ -51,6 +51,7 @@ const { getScope } = require('./scope')
 
 const HTML_ELEMENT_NAMES = new Set(require('./html-elements.json'))
 const SVG_ELEMENT_NAMES = new Set(require('./svg-elements.json'))
+const MATH_ELEMENT_NAMES = new Set(require('./math-elements.json'))
 const VOID_ELEMENT_NAMES = new Set(require('./void-elements.json'))
 const VUE2_BUILTIN_COMPONENT_NAMES = new Set(
   require('./vue2-builtin-components')
@@ -948,6 +949,8 @@ module.exports = {
         !this.isHtmlWellKnownElementName(node.rawName)) ||
       (this.isSvgElementNode(node) &&
         !this.isSvgWellKnownElementName(node.rawName)) ||
+      (this.isMathElementNode(node) &&
+        !this.isMathWellKnownElementName(node.rawName)) ||
       hasAttribute(node, 'is') ||
       hasDirective(node, 'bind', 'is') ||
       hasDirective(node, 'is')
@@ -977,7 +980,7 @@ module.exports = {
    * @param {VElement} node The node to check.
    * @returns {boolean} `true` if the node is a MathML element.
    */
-  isMathMLElementNode(node) {
+  isMathElementNode(node) {
     return node.namespace === NS.MathML
   },
 
@@ -999,6 +1002,15 @@ module.exports = {
     return SVG_ELEMENT_NAMES.has(name)
   },
 
+  /**
+   * Check whether the given name is a well-known MathML element or not.
+   * @param {string} name The name to check.
+   * @returns {boolean} `true` if the name is a well-known MathML element name.
+   */
+  isMathWellKnownElementName(name) {
+    return MATH_ELEMENT_NAMES.has(name)
+  },
+
   /**
    * Check whether the given name is a void element name or not.
    * @param {string} name The name to check.
@@ -3121,7 +3133,7 @@ function getWithDefaultsProps(node) {
 
   for (const prop of param.properties) {
     if (prop.type !== 'Property') {
-      return {}
+      continue
     }
     const name = getStaticPropertyName(prop)
     if (name != null) {
diff --git a/lib/utils/math-elements.json b/lib/utils/math-elements.json
new file mode 100644
index 000000000..46ac91ac5
--- /dev/null
+++ b/lib/utils/math-elements.json
@@ -0,0 +1,34 @@
+[
+  "math",
+  "maction",
+  "annotation",
+  "annotation-xml",
+  "menclose",
+  "merror",
+  "mfenced",
+  "mfrac",
+  "mi",
+  "mmultiscripts",
+  "mn",
+  "mo",
+  "mover",
+  "mpadded",
+  "mphantom",
+  "mprescripts",
+  "mroot",
+  "mrow",
+  "ms",
+  "semantics",
+  "mspace",
+  "msqrt",
+  "mstyle",
+  "msub",
+  "msup",
+  "msubsup",
+  "mtable",
+  "mtd",
+  "mtext",
+  "mtr",
+  "munder",
+  "munderover"
+]
diff --git a/package.json b/package.json
index ee2e37c71..b57a57e83 100644
--- a/package.json
+++ b/package.json
@@ -1,6 +1,6 @@
 {
   "name": "eslint-plugin-vue",
-  "version": "9.25.0",
+  "version": "9.26.0",
   "description": "Official ESLint plugin for Vue.js",
   "main": "lib/index.js",
   "scripts": {
diff --git a/tests/lib/rules/require-default-prop.js b/tests/lib/rules/require-default-prop.js
index 97aea4919..834f319aa 100644
--- a/tests/lib/rules/require-default-prop.js
+++ b/tests/lib/rules/require-default-prop.js
@@ -329,6 +329,28 @@ ruleTester.run('require-default-prop', rule, {
         ...languageOptions,
         parserOptions: { parser: require.resolve('@typescript-eslint/parser') }
       }
+    },
+    {
+      filename: 'test.vue',
+      code: `
+      <script setup lang="ts">
+      const defaultProps = {
+        foo: 'foo',
+      }
+      withDefaults(defineProps<{
+        foo: string;
+        bar?: number;
+      }>(), {
+        ...defaultProps,
+        bar: 42,
+      })
+      </script>
+      `,
+      languageOptions: {
+        parser: require('vue-eslint-parser'),
+        ...languageOptions,
+        parserOptions: { parser: require.resolve('@typescript-eslint/parser') }
+      }
     }
   ],
 
@@ -544,6 +566,33 @@ ruleTester.run('require-default-prop', rule, {
         ...languageOptions
       }
     },
+    {
+      filename: 'test.vue',
+      code: `
+      <script setup lang="ts">
+      const defaultProps = {
+        foo: 'foo',
+      }
+      withDefaults(defineProps<{
+        foo: string;
+        bar?: number;
+      }>(), {
+        ...defaultProps,
+      })
+      </script>
+      `,
+      errors: [
+        {
+          message: "Prop 'bar' requires default value to be set.",
+          line: 8
+        }
+      ],
+      languageOptions: {
+        parser: require('vue-eslint-parser'),
+        ...languageOptions,
+        parserOptions: { parser: require.resolve('@typescript-eslint/parser') }
+      }
+    },
     ...(semver.lt(
       require('@typescript-eslint/parser/package.json').version,
       '4.0.0'
diff --git a/tests/lib/rules/require-typed-ref.js b/tests/lib/rules/require-typed-ref.js
index 6e8797943..5b18bcb3c 100644
--- a/tests/lib/rules/require-typed-ref.js
+++ b/tests/lib/rules/require-typed-ref.js
@@ -86,6 +86,20 @@ tester.run('require-typed-ref', rule, {
       `,
       languageOptions: { parser: require('vue-eslint-parser') }
     },
+    {
+      filename: 'test.vue',
+      code: `
+        <script>
+          import { ref } from 'vue'
+          export default {
+            setup() {
+              const count = ref()
+            }
+          }
+        </script>
+      `,
+      languageOptions: { parser: require('vue-eslint-parser') }
+    },
     {
       filename: 'test.js',
       code: `
@@ -217,6 +231,30 @@ tester.run('require-typed-ref', rule, {
       ],
       languageOptions: { parser: require('vue-eslint-parser') }
     },
+    {
+      filename: 'test.vue',
+      code: `
+        <script lang="ts">
+          import { ref } from 'vue'
+          export default {
+            setup() {
+              const count = ref()
+            }
+          }
+        </script>
+      }
+      `,
+      errors: [
+        {
+          messageId: 'noType',
+          line: 6,
+          column: 29,
+          endLine: 6,
+          endColumn: 34
+        }
+      ],
+      languageOptions: { parser: require('vue-eslint-parser') }
+    },
     {
       filename: 'test.ts',
       code: `
diff --git a/tools/update-vue3-export-names.js b/tools/update-vue3-export-names.js
index e86d77c93..37e07e03d 100644
--- a/tools/update-vue3-export-names.js
+++ b/tools/update-vue3-export-names.js
@@ -158,6 +158,7 @@ function httpGet(url) {
             baseUrl.pathname = redirectUrl
             redirectUrl = String(baseUrl)
           }
+          res.destroy()
           resolve(httpGet(redirectUrl))
           return
         }