MySQL:
extracting the whole database:
sed -n '/^-- Current Database: `databaseName`/,/^-- Current Database: `/p' /path/to/the/fill/dump.sql > output.sql
single table:
sed -n '/^CREATE TABLE `tableName`/,/^\-\- Table structure for table /p' /path/to/the/fill/dump.sql > output.sql
[via this blog]
PostgreSQL:
pv dump.sql | perl -ne 'print if /^COPY public.name_of_the_table/ .. /^\\\.$/' > out.sql