No single plug-connected item is permitted to draw more than 80% of the rating of the circuit. That would be 12 amps for a 15 amp circuit and 16 amps for a 20 amp circuit. You will not find item sold with a 15 amp plug that draws more than 12 amps or an item with a 20 amp plug that draws more than 16 amps. However, there is nothing to prevent 15 or 20 amps of load from being connected through a power strip. Also, 15 amp outlets are permitted to be used on a 20 amp circuit. That would indicate that each individual 15 amp outlet or pair would need to be capable of 20 amps.
What is permitted is stated in the National Electrical Code, article 210. The use of 15 amp receptacles on a 20 amp circuit covered by Table 210.21(B)(1) and the 80% limitation is covered by paragraph 210.23(A)(1).
Re answer by Bryce
The main answer is direct and to the point, but the last part is not true.