Hydrogen Peroxide (3% chemist variety or 6% hair bleach), will provide oxygen for laundry bleaching.
Sodium Perborate - The "hot water" oxygen bleach, will bleach but requires 140F water temps and above to really become effective. However modern bleach activators such as TAED, allow bleaching action to take place at 100F. Sodium perborate is difficult to rinse and has been banned by the EU for laundry purposes because of the danger boron poses to plant and aquatic life.
Sodium Percarbonate - The "cold water" oxygen bleach, will bleach in cold water as well as hot or boiling, thus has a wider reaction time and use than sodium perborate. Sodium percarbonate releases more oxygen in wash water than sodium perborate and according to some studies than liquid hydrogen peroxide.
All oxygen bleaches will work to some extent in cold temperatures, but bleaching action increases as temperatures rise. IIRC the numbers are something like a 20% increase in bleaching power for each 10 degrees of water temperature. This is why "colour safe" bleaches were traditionally sodium perborate as it wouldn't bleach using the normal wash temperatures for colours (warm or cool water). However if one washed those same colours at 140F or above, that is a different story.
Sodium perborate is somewhat inexpensive, which was the reason it was used on both sides of the pond. European laundry products have begun phasing the stuff out (Persil/Germany now contains sodium percarbonate bleaching agents), while American laundry detergents/bleaches mostly are sodium perborate (Biz, Clorox II dry), though most of the "Oxi" types are sodium percarbonate.
In general chlorine bleach is more effective in removing a wider array of stains and at lower temperatures/contact times than oxygen bleaches. However due to the damage repeated and or improper use of LCB, many prefer to use oxygen bleach. To get the whitening power of chlorine bleach, oxygen bleaches require either hot to boiling water or long contact times.
Oxygen bleaches and chlorine bleaches cancel each other out, one can be used to neutralise the other. That is you can pre-treat a bad stain with LCB, then launder with an oxygen bleach. The later will cancel out the former. White vinegar will also remove remaining traces of chlorine bleach.