diff --git a/packages/babel-parser/src/tokenizer/index.js b/packages/babel-parser/src/tokenizer/index.js
index 1eefa40ade01..96ff0c21297a 100644
--- a/packages/babel-parser/src/tokenizer/index.js
+++ b/packages/babel-parser/src/tokenizer/index.js
@@ -113,6 +113,7 @@ export default class Tokenizer extends LocationParser {
   // parser/util.js
   /*::
   +unexpected: (pos?: ?number, messageOrType?: string | TokenType) => empty;
+  +expectPlugin: (name: string, pos?: ?number) => true;
   */
 
   isLookahead: boolean;
@@ -405,10 +406,14 @@ export default class Tokenizer extends LocationParser {
     }
 
     if (
-      this.hasPlugin("recordAndTuple") &&
-      (next === charCodes.leftCurlyBrace ||
-        next === charCodes.leftSquareBracket)
+      next === charCodes.leftCurlyBrace ||
+      (next === charCodes.leftSquareBracket && this.hasPlugin("recordAndTuple"))
     ) {
+      // When we see `#{`, it is likely to be a hash record.
+      // However we don't yell at `#[` since users may intend to use "computed private fields",
+      // which is not allowed in the spec. Throwing expecting recordAndTuple is
+      // misleading
+      this.expectPlugin("recordAndTuple");
       if (this.getPluginOption("recordAndTuple", "syntaxType") !== "hash") {
         throw this.raise(
           this.state.pos,
@@ -426,14 +431,8 @@ export default class Tokenizer extends LocationParser {
         this.finishToken(tt.bracketHashL);
       }
       this.state.pos += 2;
-    } else if (
-      this.hasPlugin("classPrivateProperties") ||
-      this.hasPlugin("classPrivateMethods") ||
-      this.getPluginOption("pipelineOperator", "proposal") === "smart"
-    ) {
-      this.finishOp(tt.hash, 1);
     } else {
-      throw this.raise(this.state.pos, Errors.InvalidOrUnexpectedToken, "#");
+      this.finishOp(tt.hash, 1);
     }
   }
 
@@ -1080,8 +1079,13 @@ export default class Tokenizer extends LocationParser {
     if (val == null) {
       this.raise(this.state.start + 2, Errors.InvalidDigit, radix);
     }
+    const next = this.input.charCodeAt(this.state.pos);
 
-    if (this.input.charCodeAt(this.state.pos) === charCodes.lowercaseN) {
+    if (next === charCodes.underscore) {
+      this.expectPlugin("numericSeparator", this.state.pos);
+    }
+
+    if (next === charCodes.lowercaseN) {
       ++this.state.pos;
       isBigInt = true;
     }
@@ -1154,6 +1158,10 @@ export default class Tokenizer extends LocationParser {
       }
     }
 
+    if (next === charCodes.underscore) {
+      this.expectPlugin("numericSeparator", this.state.pos);
+    }
+
     if (next === charCodes.lowercaseN) {
       // disallow floats, legacy octal syntax and non octal decimals
       // new style octal ("0o") is handled in this.readRadixNumber
diff --git a/packages/babel-parser/test/fixtures/esprima/invalid-syntax/migrated_0065/options.json b/packages/babel-parser/test/fixtures/esprima/invalid-syntax/migrated_0065/options.json
index 895412e4e5f7..ff1ac8247e68 100644
--- a/packages/babel-parser/test/fixtures/esprima/invalid-syntax/migrated_0065/options.json
+++ b/packages/babel-parser/test/fixtures/esprima/invalid-syntax/migrated_0065/options.json
@@ -1,3 +1,3 @@
 {
-  "throws": "Unexpected character '#' (1:2)"
-}
+  "throws": "Unexpected token, expected \";\" (1:2)"
+}
\ No newline at end of file
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/input.js b/packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/input.js
deleted file mode 100644
index 7a5a058e169c..000000000000
--- a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/input.js
+++ /dev/null
@@ -1 +0,0 @@
-1n
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.class-private-properties/options.json b/packages/babel-parser/test/fixtures/experimental/_no-plugin/.class-private-properties/options.json
deleted file mode 100644
index 85d7fee65f66..000000000000
--- a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.class-private-properties/options.json
+++ /dev/null
@@ -1,4 +0,0 @@
-{
-  "throws": "This experimental syntax requires enabling the parser plugin: 'classPrivateProperties' (2:3)",
-  "plugins": []
-}
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.class-private-properties/input.js b/packages/babel-parser/test/fixtures/experimental/_no-plugin/class-private-properties/input.js
similarity index 100%
rename from packages/babel-parser/test/fixtures/experimental/_no-plugin/.class-private-properties/input.js
rename to packages/babel-parser/test/fixtures/experimental/_no-plugin/class-private-properties/input.js
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/class-private-properties/options.json b/packages/babel-parser/test/fixtures/experimental/_no-plugin/class-private-properties/options.json
new file mode 100644
index 000000000000..6904d075e9f1
--- /dev/null
+++ b/packages/babel-parser/test/fixtures/experimental/_no-plugin/class-private-properties/options.json
@@ -0,0 +1,3 @@
+{
+  "throws": "This experimental syntax requires enabling one of the following parser plugin(s): 'classPrivateProperties, classPrivateMethods' (2:2)"
+}
\ No newline at end of file
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/input.js b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/input.js
new file mode 100644
index 000000000000..0fa2562844bb
--- /dev/null
+++ b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/input.js
@@ -0,0 +1 @@
+0b1_0
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/options.json b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/options.json
similarity index 58%
rename from packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/options.json
rename to packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/options.json
index 8e44fabed2f5..180f7a692d48 100644
--- a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.bigint/options.json
+++ b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator-radix/options.json
@@ -1,4 +1,3 @@
 {
-  "throws": "This experimental syntax requires enabling the parser plugin: 'bigInt' (1:1)",
-  "plugins": []
-}
+  "throws": "This experimental syntax requires enabling the parser plugin: 'numericSeparator' (1:3)"
+}
\ No newline at end of file
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.numeric-separator/input.js b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator/input.js
similarity index 100%
rename from packages/babel-parser/test/fixtures/experimental/_no-plugin/.numeric-separator/input.js
rename to packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator/input.js
diff --git a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.numeric-separator/options.json b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator/options.json
similarity index 53%
rename from packages/babel-parser/test/fixtures/experimental/_no-plugin/.numeric-separator/options.json
rename to packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator/options.json
index 6996ba6f1573..ed83acd92c8c 100644
--- a/packages/babel-parser/test/fixtures/experimental/_no-plugin/.numeric-separator/options.json
+++ b/packages/babel-parser/test/fixtures/experimental/_no-plugin/numeric-separator/options.json
@@ -1,4 +1,3 @@
 {
-  "throws": "This experimental syntax requires enabling the parser plugin: 'numericSeparator' (1:17)",
-  "plugins": []
-}
+  "throws": "This experimental syntax requires enabling the parser plugin: 'numericSeparator' (1:1)"
+}
\ No newline at end of file
diff --git a/packages/babel-parser/test/fixtures/experimental/record-and-tuple/not-enabled-hash/options.json b/packages/babel-parser/test/fixtures/experimental/record-and-tuple/not-enabled-hash/options.json
index 52f524480cd0..9565d05f7929 100644
--- a/packages/babel-parser/test/fixtures/experimental/record-and-tuple/not-enabled-hash/options.json
+++ b/packages/babel-parser/test/fixtures/experimental/record-and-tuple/not-enabled-hash/options.json
@@ -1,4 +1,3 @@
 {
-  "plugins": [],
-  "throws": "Unexpected character '#' (1:0)"
-}
+  "throws": "This experimental syntax requires enabling the parser plugin: 'recordAndTuple' (1:0)"
+}
\ No newline at end of file