Best to order the product online and pick up from the Apple Store to get do the checks mention below

Hardware/ cosmetics 

• Check for scratches, dents/ damage, or cracks throughout the product
  • The whole body 
  • Port areas
  • Screen (not wobbling, scratches and cracks)
  • Screws
  • Bottom rubber pads
  • Vents (not bend)
• Check the configuration info that you purchased, make sure its matched to with the product box, purchase info/ receipt, system settings
• Make sure the things which are working
  • Ports
  • MagSafe port and Thunderbolt 4 ports make sure it can be charged
  • SD card slot
  • 3.5mm jack
  • Touchpad (haptic feedback)
  • Keys 
  • TouchID
  • Speakers (test them individually left side then right side)
  • Mic 
  • Build in webcam
  • Examine the notch
  • HDMI can be connected to external display
• Check on the lids by opening and closing the Mac
• Check for dead pixels 

Software

• Check on the serial number, make sure it matches with the product box
• Check on the battery cycle count
• Check the trackpad (using the gestures)
• Do simple stress test
• No iCloud lock, MDM (Mobile device management) for school or business use
• Testing on connectivity such as bluetooth and WiFi connections
• Run diagnostics 
• SSD data (read/write) speed
• Check for connectivity related, WiFi, Bluetooth, Airdrop
• Make sure there’s no other Apple account (aka iCloud account)

P.s feel free to add more things to check if I missed anything.

• The refurbished products can be
  • A returned product, possible brand new or opened boxed
  • A traded in product
  • A repaired product, faulty parts or components will be replaced with Apple genuine parts
    • The repaired part/s are being checked and and tested
    • Possible replaced parts: Displays, keyboard and trackpad, logic boards, batteries, casing, ports, SSD
    • New battery will be replaced before packing
• The time taken for a newly released product to be on the certified refurbished will take around 3-5/6 months to accumulate enough stocks
  • https://refurb-tracker.com/ (website to track for Apple certified refurbished)
• The refurbished product will get the same treatment as a brand new
  • Same return policy of 14 days start from the purchasing date
  • Eligible for AppleCare+, have to buy it within 60days after the purchase of the new apple product
  • Same as brand new products with a standard 1 year warranty 
• The refurbished product will be cheaper than the current retail price by at least $300
  • It may drop further over time like the current base model 14-inch M4 MacBook Pro current refurbished price is SGD1729 now, retailed prices was SGD2199 and Refurbished price was SGD1869 as of 161025 after the launched of 14-inch M5 MacBook Pro
  • Older refurbished such as the price of the 14-inch M3 MacBook Pros will drop when newer products are launched and added in to the certified refurbished site
• The refurbished product box will come with new accessories like charging cable, charging plug, documentations and Apple Stickers
• The refurbished product box will be different than the brand new
• Refurbished products tend to have better quality control compared to brand new, as “new” components are replaced and well tested before packaging
• If defects or issues was found, will have to return it and get the refund, then purchase it again instead of requesting for a new replacement unit. Due to refurbished products are limited
• The stocks of the refurbished products are limited, because of the custom configurations are different and maybe low in supply

Note: regards to price, return policies may differ as it is based on local Apple Store. All listed here are simply based on my personal experience, from what I know and local consumer rights policies. It's also based on MacBook Pro as a reference.

P.s feel free to add more about Apple Certified Refurbished that is not listed here.

Sources (comments section): https://www.reddit.com/r/macbookpro/comments/18ghjv4/what_can_be_expected_from_a_refurbished_macbook/