Why no to carrots, miss Wiggle? I use that on a regular basis and have never had any problems. The only thing is, my livebearers find it a bit chewy, so it's mainly for the bristlies. I have used apple (and pear and banana) occasionally, but wouldn't do so on a regular basis, as it is a bit acid; would probably pass on the grapes for this reason.
Other veggies I use include peas (the no. 1 favourite!), courgette (zucchini), cucumber, spinach, chard, lettuce, broad beans, broccoli and (as from yesterday) pumpkin. How often depends on what fish you have: with the livebearers I feed veg three times a week, but by now most of my tanks contain a flock of hungry bristlenose juvies, so that means veggies every day.
IME boiled zucchini will sink of its own accord but cucumber won't. Boiled peas sink; I mush them up a bit so there are lots of tiny fragments for my fry. Lettuce I weigh down with bits of plant weight.