MySQL grammar: Apache Doris, Apache Impala.
PostgreSQL grammar: Greenplum.
Non standard SQL: Clickhouse, Apache Druid.
A third standard, more common than either MySQL or PostgreSQL, is Apache Calcite, which is the SQL dialect used by a wide range of Apache projects (Apex, Beam, Drill, Druid, Flink, Hive, Kylin, Phoenix, Pinot, Samza, Storm), Alibaba MaxCompute, Dremio, OmniSci, Qubole, SQLl-Gremlin, Tencent Big Data Suite, and others.
Teradata and Google BigQuery use their own SQL dialects.