diff --git a/changelog.d/20100.bugfix b/changelog.d/20100.bugfix
new file mode 100644
index 0000000000..8f488be0bf
--- /dev/null
+++ b/changelog.d/20100.bugfix
@@ -0,0 +1 @@
+Fix schema deltas and background updates containing a literal `%` failing on PostgreSQL. Broken since Synapse v1.82.0.
diff --git a/synapse/storage/database.py b/synapse/storage/database.py
index 023014276b..85a0e6ea93 100644
--- a/synapse/storage/database.py
+++ b/synapse/storage/database.py
@@ -462,6 +462,17 @@ class LoggingTransaction:
         )
 
     def execute(self, sql: str, parameters: SQLQueryParameters = ()) -> None:
+        if not parameters:
+            # Don't hand an empty parameter collection to the driver.
+            #
+            # psycopg2 performs `%`-substitution on the SQL whenever it is given
+            # *any* parameters, including an empty collection, so a statement
+            # containing a literal `%` (e.g. `LIKE 'foo%'`) fails with
+            # `IndexError: tuple index out of range`. Omitting the parameters
+            # altogether disables the substitution.
+            self._do_execute(self.txn.execute, sql)
+            return
+
         self._do_execute(self.txn.execute, sql, parameters)
 
     def executemany(self, sql: str, *args: Any) -> None:
diff --git a/tests/storage/test_database.py b/tests/storage/test_database.py
index 6213abd753..82d0e4f5aa 100644
--- a/tests/storage/test_database.py
+++ b/tests/storage/test_database.py
@@ -142,6 +142,42 @@ class ExecuteScriptTestCase(unittest.HomeserverTestCase):
         )
 
 
+class LiteralPercentTestCase(unittest.HomeserverTestCase):
+    """Tests that SQL containing a literal `%` can be run without parameters.
+
+    psycopg2 `%`-substitutes the SQL it is given whenever any parameters are
+    supplied, so passing it an empty collection of parameters makes it choke on
+    statements like `LIKE 'foo%'`. Several schema deltas and background updates
+    contain such statements.
+    """
+
+    def prepare(self, reactor: MemoryReactor, clock: Clock, hs: HomeServer) -> None:
+        self.db_pool: DatabasePool = hs.get_datastores().main.db_pool
+
+    def test_execute(self) -> None:
+        """A literal `%` survives `LoggingTransaction.execute`."""
+
+        def run(txn: LoggingTransaction) -> None:
+            txn.execute("SELECT 'a/b' LIKE 'a/%'")
+            self.assertEqual(txn.fetchall(), [(True,)])
+
+        self.get_success(self.db_pool.runInteraction("test_execute", run))
+
+    def test_executescript(self) -> None:
+        """A literal `%` survives `BaseDatabaseEngine.executescript`.
+
+        This is the path taken by `.sql` schema deltas.
+        """
+
+        def run(conn: LoggingDatabaseConnection) -> None:
+            cur = conn.cursor(txn_name="test_executescript")
+            self.db_pool.engine.executescript(
+                cur, "CREATE TABLE percent_test (name TEXT); SELECT 'a/b' LIKE 'a/%';"
+            )
+
+        self.get_success(self.db_pool.runWithConnection(run))
+
+
 @attr.s(slots=True, auto_attribs=True)
 class TransactionMocks:
     after_callback: Mock