Some popular PostgreSQL clients will send SQLs that cause exceptions:

pgAdmin

• SET client_min_messages=notice
• SELECT set_config('bytea_output','escape',false) FROM pg_settings WHERE name = 'bytea_output'
• SET client_encoding='UNICODE'
• SELECT version(), should return column header version but not public.version(), since pgAdmin reads row['version'], see connection.py L470
• SELECT ... FROM pg_database db WHERE db.datname = current_database() (function current_database())
• SELECT ..., rolsuper ... FROM pg_catalog.pg_roles WHERE rolname = current_user (column pg_roles.rolsuper)
• SELECT ... FROM pg_database db LEFT OUTER JOIN pg_tablespace ta ON db.dattablespace = ta.oid WHERE db.oid > 100000::OID ORDER BY datname, got Data conversion error converting "INTEGER to BLOB" (::OID should be supported, and this query should return at least one row, otherwise pgAdmin cannot open SQL window)
• SELECT ..., has_schema_privilege(nsp.oid, 'CREATE'), ... FROM pg_namespace nsp WHERE ... (function has_schema_privilege())
• SELECT format_type(23, NULL), should return 'integer' but not null
• SET LOCAL join_collapse_limit=8 (SESSION | LOCAL not supported)

HeidiSQL

• SHOW ssl
• SET search_path TO 'public', '$user' (single quote not supported)
• SELECT *, NULL AS data_length, pg_relation_size(QUOTE_IDENT(t.TABLE_SCHEMA) || '.' || QUOTE_IDENT(t.TABLE_NAME))::bigint AS index_length, c.reltuples, obj_description(c.oid) AS comment FROM "information_schema"."tables" ... (function pg_relation_size(), obj_description())
• SELECT DISTINCT a.attname AS column_name, ... FROM pg_attribute AS a JOIN pg_class AS pgc ON pgc.oid = a.attrelid WHERE ... AND pg_table_is_visible(pgc.oid) ... (function pg_table_is_visible())
• SELECT "p"."proname", "p"."proargtypes" FROM "pg_catalog"."pg_namespace" ...
• SELECT ... FROM "pg_constraint" AS "c" ... (table pg_constraint)
• SHOW ALL

DBeaver

• SELECT current_schema(),session_user (function session_user())
• SELECT t.oid,t.*,c.relkind FROM pg_catalog.pg_type t LEFT OUTER JOIN pg_class c ON c.oid=t.typrelid WHERE typnamespace=$1 (column pg_type.typrelid)
• SELECT ..., pg_get_expr(ad.adbin, ad.adrelid, true) ... FROM pg_catalog.pg_attribute a ... LEFT OUTER JOIN pg_catalog.pg_attrdef ad ON ... (function pg_get_expr needs to allow 3rd optional args)
• SHOW search_path
• SELECT ..., pg_catalog.pg_stat_get_numscans(i.indexrelid) ... FROM pg_catalog.pg_index i ... (function pg_stat_get_numscans)
• SELECT ... FROM pg_catalog.pg_constraint c ... WHERE c.conrelid=$1 ORDER BY c.oid (table pg_constraint)
• SELECT ... FROM pg_catalog.pg_type t JOIN pg_catalog.pg_namespace n ON t.typnamespace = n.oid WHERE ... n.nspname = ANY (current_schemas(true)) ... (function current_schemas)
• Above SQL (expression = ANY(ARRAY))

If H2 supports these features, these popular SQL clients can access H2 databases easily.