What kind of fish oil supplement should I take that doesn't have any taste/odor?

This doesn’t always work, but I use the ones that have a lemon flavor and promise not to have any aftertaste. The ones with lemon and the “no aftertaste” ones usually really don’t have an aftertaste but the keyword is ‘usually.’ Sometimes there still is a fishy burp about a hour later. Another option is to take the 3-in-one-Omega 3 pills (fish, flax and borage oil), again with a lemon flavor. It’s still a 80–90% hit-miss.
The fish-burps still are not as bad as the Omega 3 peanutbutter I bought—first few dabs… perfectly fine Jif-like taste, then one day I took a bite of my toast with pnb, and…fish. Bit again… FISH. what the… sniffed the jar and it was a weird PNB+fish odor. GAG. Never again. Those Omega-3’s are not worth forcing an acquired taste for breaking my fast with PNB+fish. Beware!!!
IF you find a perfect fish oil pill, let me know!

Try Flax seed oil instead, if you are trying to get Omega-3 oil. Three capsules a day morning, noon, and night.

A tough one for sure.
Keep it in the refrigerator to help minimize the taste when taking it.
The better the fish oil, the less the bad breath/burp reactions.
Your fish oil must list the specific species of fish used to make the oil. Salmon, tuna, sardines and anchovies are the richest sources.
Your fish oil must contain omega 3 fatty acids – not just “fish oil”. They are not the same thing.
It should list the total amount of EPA and DHA on the label. This should add up to the total amount of oils in the product. If they don’t, you’re being sold a bunch of filler oils (yuck!)
The oil should be pressed from only the flesh of the fish. Don’t accept products that are pressed from heads, tails or internal organs.
The fish used for the oil should be health screened and disease free.
Your fish oil should be guaranteed to be 100% pure.

Many brands do not do that. Just experiment or look for ones that advertise that they don’t.

I take one that’s a gel cap and have never had any fish burps. I think it’s Nature’s Way, or something like that. It’s a green bottle with a yellow label. I can look if you’d like.I take three or four caps after dinner and no problems at all.

“cold pressed” salmon oil has a less fishy taste and refrigerating the capsules does delay the gelatin dissolving so you get less chance of “repeating” fish.

If you are taking it for the purpose of getting Omega 3 fatty acids, you can take Flax seed oil, or you can add ground (not whole, because it goes right through you un-digested) flax seed to your food. I think Borage oil is another good choice, but don’t quote me on that, check it out of Google.

I have read that fish oil taste is because the fish oil is no longer fresh. I started taking Nordic Naturals when I was pregnant because they are a pharmaceutical grade, which means mercury and other such things have been removed from them. I highly recommend them. I have never had a fishy burp and it takes me about three months to go through a bottle. They are a little more expensive, but it’s worth it, especially because you’re also guaranteed to be exposed to less mercury. :)