Hi Jemima, I would buy the strongest available - it will remain potent the longest. In a laboratory setting, we typically use 30 - 35%, and it contains an additive that stabilizes it. We also store it in the refrigerator. I would try making a small batch of the cleaning solution, and test it in several out-of-sight areas (like under the kitchen sink!). It could take over night to see damage (e.g. bleaching of your wood) so be patient.
You could even make a dilution series, and test several different items (wood, plastic, metal) with each concentration, but I'm clearly turning this into a science fair project (sorry, I just cannot help myself). Good luck - and I'd be interested to hear how it works.

I would recommend sticking with the peroxide that you can buy over the counter, I think typically around 3-5%. Hydrogen peroxide is a very strong oxidizing agent. While it is safe in lower concentrations, in higher concentrations it can be quite dangerous.

There are a variety of concentrations out there (I believe hospitals use a slightly higher concentration than what is for sale at the pharmacy, and as lazy_lurker indicated, labs use solutions in the 30% range), they are typically not available to the general public. If you do somehow find your hands on anything other than that which you can get at your pharmacy, please don't use it. Unless you know what you are doing, using higher concentrations of hydrogen peroxide can lead to explosions or spontaneous combustion.

unstabilized H202 starts at 35% concentrations. Ive made my own 3% dilution for years and use it for everything, really. Counter tops, laundry, stain fighting, gardening, health concoctions etc. Ive heard others using dilutions as high as 6% for household use but personally I wouldn't go any higher than that.
As wonderful as H202 is I have to caution anyone that chooses to buy 35% or higher H202, it is caustic and can cause burns. You must take proper precaution and handle with care, including the use of safety glasses and gloves.