The culprit is not the LEDs themselves, they're harmless, but the TLC5940s, which switch at high frequency to control LED brightness through PWM. You can't filter the PWM outputs (you can, but then the brightness control doesn't work properly anymore), so that's out, but you can try to do something about decoupling power supplies. Not guaranteed to work, the fact that the scope's probe picks up the signal unconnected indicates that it's probably radiated, but it's worth trying.
Decouple the TLC5940s properly. They have to provide a lot of power, so that means 100\$\mu\$F, 1\$\mu\$F and 100nF all parallel on the power supply for each device, the smallest value closest to the pins.
Decouple your 5V power to the microcontroller properly: 100nF close to the pins.
The problem is common to this type of audio system. I would bet if you looked at the noise spectrum you would see 60 Hz plus many of the harmonic frequencies (120 Hz, 180 Hz, 240 Hz, etc.). The fact that it is more than just 60 Hz, or 50 Hz in some countries, is an indicator that it is not just simple ground loops.
I would also bet that your laptop power supply has only a 2-prong AC plug-- lacking the third ground plug.
In this type of power supply, the output is electrically isolated from the AC input. But it is not perfectly isolated. There is a small amount of current that flows between the isolation barrier. This is called the "leakage current". It is not a lot of current, but it doesn't have to be.
Some laptop users report getting shocked or having a tingling sensation in the legs when using the laptop while wearing shorts! The reason for this is that leakage current is going through the screws in the bottom of the laptop and into their legs. It sounds dangerous, but the amount of current is well below the safety limit. It is more startling than anything else. If you are wearing pants then you're insulated.
Laptop chargers that have the 3rd prong on the AC plug do not have this problem because that third plug is connecting the laptop chassis shield to ground-- forcing that leakage current to go to ground instead of into your leg. Of course, there is no leakage if you are running off of batteries.
In your case, the leakage current is not just going into your leg, but into your radio receiver. The solution to this is to properly ground your laptop.
You will have to experiment with this a little bit to find the best solution. Getting a power supply with a 3-prong AC plug is the best, but not always possible. The next option is to find something on your laptop that you can ground. Make an adapter from that 3rd prong to "something". That something could be the signal-ground on the output cable of your power supply. It could be a screw on the laptop. Or a shield on an unused laptop connector. Or the ground/shield on your audio cable.
Make that 3rd prong adapter, but leave the other end bare for the moment. Then start poking it around to see if or where you can connect it and have the noise go away. Once you have found a place or two, then finish up the adapter so it is easy to use.
Two warnings when doing this: Make sure that whatever you are grounding is actually ground! On the power supply output, make sure you ground the negative or gnd conductor. And when poking around, understand that you might actually have to poke a little hard. Both the bare wire and whatever you are poking will likely have a thin layer of non-conductive stuff on it, and you need to apply enough force to poke through it. Rubbing sometimes helps too. The non-conductive layer is sometimes paint on screws, or an oxide (rust) on the metals.
Oops, here is a 3rd warning: Be super careful when making that 3rd prong adapter. You're messing with potentially lethal voltages and we don't want you to die. Build the adapter in a way that there is no possibility of it failing and shorting out against either one of the other two conductors in the AC plug.
Give it a try and report back what you found!
Best Answer
Most of the noise (perhaps 1-2mVp-p) will be due to the reference, so putting a capacitor on the op-amp non-inverting input (with a series resistor in the case of the TL071) makes some sense. 100nF with a 20K series resistor (about 22-23K equivalent total) will give you a cutoff frequency of around 71Hz if I did the math right. Similar to @WhatRoughBeast's comment but with added series resistance.
On the other hand if you use a lower noise reference such as an LM4040-2.048 or -2.5 which has something like 250uVp-p noise typically, the capacitor and resistor may not be necessary, since the amplifier is contributing significantly.