Actually set index already woks for everything. Here are some tests.

dbms/src/Storages/MergeTree/MergeTreeIndexSet.cpp

@@ -411,7 +411,10 @@ static bool checkAtomName(const String & name)
             "greaterOrEquals",
             "in",
             "notIn",
-            "like"
+            "like",
+            "startsWith",
+            "endsWith",
+            "multiSearchAny"
             };
     return atoms.find(name) != atoms.end();
 }

dbms/tests/queries/0_stateless/00965_set_index_string_functions.reference

@@ -0,0 +1,16 @@
+9	abra
+14	abracadabra
+		"rows_read": 4,
+9	abra
+10	cadabra
+11	crabacadabra
+14	abracadabra
+15	cadabraabra
+		"rows_read": 6,
+9	abra
+14	abracadabra
+		"rows_read": 4,
+1	ClickHouse is a column-oriented database management system (DBMS)
+2	column-oriented database management system
+13	basement
+		"rows_read": 6,

dbms/tests/queries/0_stateless/00965_set_index_string_functions.sh

@@ -0,0 +1,66 @@
+#!/usr/bin/env bash
+
+CURDIR=$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)
+. $CURDIR/../shell_config.sh
+
+$CLICKHOUSE_CLIENT --query="DROP TABLE IF EXISTS set_idx;"
+
+$CLICKHOUSE_CLIENT -n --query="
+SET allow_experimental_data_skipping_indices = 1;
+CREATE TABLE set_idx
+(
+    k UInt64,
+    s String,
+    INDEX idx (s) TYPE set(2) GRANULARITY 1
+) ENGINE = MergeTree()
+ORDER BY k
+SETTINGS index_granularity = 2;"
+
+$CLICKHOUSE_CLIENT --query="INSERT INTO set_idx VALUES
+(0, 'ClickHouse - столбцовая система управления базами данных (СУБД)'),
+(1, 'ClickHouse is a column-oriented database management system (DBMS)'),
+(2, 'column-oriented database management system'),
+(3, 'columns'),
+(4, 'какая-то строка'),
+(5, 'еще строка'),
+(6, 'some string'),
+(7, 'another string'),
+(8, 'computer science'),
+(9, 'abra'),
+(10, 'cadabra'),
+(11, 'crabacadabra'),
+(12, 'crab'),
+(13, 'basement'),
+(14, 'abracadabra'),
+(15, 'cadabraabra')"
+
+# STARTS_WITH
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE startsWith(s, 'abra')"
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE startsWith(s, 'abra') FORMAT JSON" | grep "rows_read"
+
+# ENDS_WITH
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE endsWith(s, 'abra')"
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE endsWith(s, 'abra') FORMAT JSON" | grep "rows_read"
+
+# COMBINED
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE startsWith(s, 'abra') AND endsWith(s, 'abra')"
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE startsWith(s, 'abra') AND endsWith(s, 'abra') FORMAT JSON" | grep "rows_read"
+
+# MULTY_SEARCH_ANY
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE multiSearchAny(s, ['data', 'base'])"
+$CLICKHOUSE_CLIENT --query="SELECT * FROM set_idx WHERE multiSearchAny(s, ['data', 'base']) FORMAT JSON" | grep "rows_read"
+
+$CLICKHOUSE_CLIENT --query="DROP TABLE set_idx;"
+
+
+
+
+
+
+
+
+
+
+
+
+