I check the Auction House almost every day. I am heavily invested in decorating houses and like to go through the usual dozen or so pages in the decoration list for ones that might not be so common to obtain. I know my kinship appreciates if I find some non-bound things from an instance, festival, or barterer that might appeal to them. When Dragon and the Storm was constantly being ran, the dragon painting was being sold around 50g (which is actually a pretty good deal to me). Doormats, flower beds, and trees can brighten up newcomers when they aren't available all the time.
I'll also check the cosmetic pets to see if there are any that are being sold for a decent price. Especially in the past couple months when they introduced more pets onto the delving barterer without knowing when more gear would be available, those pets were being offered for some good prices for the amount of delving writs required. Sometimes, I buy a pet that requires between 100 to 250 tokens just to spend all the gold accumulated over the years. Extra stuff can be used as rewards for kinship parties and events.
Otherwise, I very rarely look for other things. Occasionally, I'll check for reputation tokens, peruse expiring items, or even check for any of the Monster Play portrait frames (if they can be listed). Since I'm a premium player, only 5 items can be placed on the Auction House for each of my characters. I sometimes throw spare crafting items but usually hold onto most items until someone specifically requests them in the Trade channel. I was able to help someone with an instance theme box one time and give someone some of the floor surface and wallpaper non-bound items from the recent expansions.