We've reported on Setting Up a USB 3.0 External PS4 HDD on Firmware 4.50, Sony's PS4 External HDD Restrictions, covered PS4 USB HDDs with PSN Accounts and shared a tutorial on PS4 Apps and Games on an External HDD and here comes some USB 3.0 250GB-8TB external hard drive recommendations that can be used with the PlayStation 4! 
Below is a summarized list derived from PS4Home.com with various pricing and size details from the manufacturers and online retailers, as follows:

Below is a summarized list derived from PS4Home.com with various pricing and size details from the manufacturers and online retailers, as follows:
- Seagate Backup Plus / Fast - available in 500GB at $49.99, 1TB at $54.99, 1.5TB at $74.00, 2TB at $79.99, 4TB at $119.00, 4TB Fast at $195.07 and 5TB at $159.99
- WD My Passport - Ultra 1 (1TB) at $59.99, Ultra 2 (2TB) at $79.99, 3TB at $109.99 and available in 4TB at $119.99
- Toshiba Canvio Basics - available in 500GB at $48.50, 1TB at $54.99, 2TB at $83.49 and 3TB at $97.35
- Seagate Expansion Portable USB HDD - available in 500GB at $49.99, 1TB at $54.99, 2TB at $79.99 and 4TB at $119.99
- Seagate Expansion USB HDD - available in 2TB at $89.99, 3TB at $99.99, 4TB at $109.99 and 5TB at $119.99
- Samsung P3 Portable - available in 500GB at $49.99, 1TB at $64.99, 2TB at $89.99 and 4TB at $179.99
- Samsung T3 Portable SSD - available in 250GB at $108.99, 500GB at $197.99, 1TB at $381.94 and 2TB at $789.95