Check the advert first
Many adverts say that if you have not heard back by a certain date, you should consider your application unsuccessful. If so, respect that. Government adverts usually say this.
When a follow-up makes sense
For private-sector roles with a named contact, a short, polite email one to two weeks after the closing date is reasonable.
What to say
Keep it to three sentences: the role and reference number, the date you applied, and your continued interest. Attach nothing new unless asked.
Track everything
Record each application, the date sent, the closing date and any follow-up. A simple spreadsheet tracker stops you missing replies or contacting the same employer twice.