(prepositions)

• under
  • lower lever/layer (spatial) : the ball rolled under the car.
  • in a situation/condition : he's been under pressure lately.
  • numbers (age/quantity) : the bar is popular among the under -40 crowd.
  • prefix : understimate, underweight

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• below
  • in relation to levels ON THE SAME PLANE :
    • 'put in the comments down below'
    • 'the number below 37 one on the list'
    • 'the apartment below'
    • 'jane is below tom in the rankings'
  • in relation to a reference point : below average or below standards

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• beneath (more formal than 'under', prettier and more sophisticated)
  • something under the SURFACE or TOUCHING it : beneath the sea or beneath my shirt
  • masked (emotions, thoughts) : 'there's a sad person beneath that happy face.'
  • level, status : he is on a level beneath mine so i don't have time to talk to him.

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• underneath (more formal and emphatic than under)

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• "he has 5 layers on under/beneath/underneath the jacket."
• "the people in the apartment underneath/below us are so noisy."
• "you might find the keys under/underneath/beneath the pile of laundry."