In no particular order: Check each supply connection again, carefully with a flashlight. Check both when you run it and when it's off. Pull out the faucet and run it - could be leaking on that end. Check the compression nut of each shut-off valve, if there's even a single drop forming, then get two wrenches to snug it up a bit more (hold with one wrench, turn with the other) (that's the nut that connects it to the supply pipe. Check the stem of each supply valve, if there's a drip, you can tighten the cap nut (nut under the handle) just a bit. BTW, if they are old "multi-turn" valves that happen to be leaking, then replace them with "1/4 turn" valves instead (much better).

Should be minimal, but if you only need the filtered water for cooking and drinking a dedicated faucet may be the way to go. You don't want to filter water that you don't have to. Lots of luck. .................................... "If all else fail, read the directions"

There are quality fittings, quality polyethylene tubing (various colors), and quality filter and RO housings that may be purchased. I have never had one leak unless I assembled it incorrectly, much less explode. (House water pressure should be regulated, if high.) I did have a transparent (cheap HW store) line to my centrifugal humidifier harden and fail after being illuminated for 10 years by my kilowatt metal halide plant light. I don't expect the black PE replacement tubing to have that issue. Notwithstanding the leakage explosion history, risk analysis suggests belt and suspenders design, so water leakage alarm/auto shut offs are called for to protect floors and "stuff" sitting on floors. The complex of devices and tubing in this photo compose most of the core pat of my RO system. This system feeds faucets at three sinks, a freezer ice maker, and said centrifugal humidifier. It has pre-filtering, of course, as well as pre-heating, and feeds storage tanks in two locations. Except for copper tubing in one stretch where damage is possible, all of the tubing is either 3/8 or 1/4 polyethylene. Fittings are John Guest or similar. Permeate pumps are ERP-500 (much quieter than ERP-1000). RO control and RO filters by SpectraPure. Heating control by bayite with further modifications. Leak alarm and shut off in various locations is by Floodstop. The point is that here and in other locations there are a lot of opportunities for tubing and fitting failures, and they haven't happened where appropriate parts are used.

Kitchen remodel happening and having trouble figuring out how to drain my water softener into my kitchen plumbing. Can I use a dishwasher style air gap? Drain into the vent stack? Dedicated plumbing under the sink? etc.