ADC distortion when writing to microSD
In the comments @Anonymous and @brhans raise an excellent point. In your question you ask how to get rid of this problem, but really you need to understand the problem better first. Is the problem digital or analog? If it's analog, is it due to power supply or other effects? Here are some troubleshooting steps I would recommend:


Examine your data more closely. Per @Anonymous and @brhans, the problem could be somewhere in the data handling. If so, I would expect that these "spikes" contain some pathological values, like 0x00, or 0xFF, or 0x7F, or 0x80. Even if it's not one of these values, any consistent or repeated values should be highly suspect since the real world contains noise. If you see any of these symptoms, then the problem exists in the digital domain.
As @brhans recommends, your next step is to tie ADC_IN to a constant DC value. If the spikes are in the data, then the problem must exist somewhere in the microcontroller. Either noise is getting in on the microcontroller's power supply, or the code is garbling the samples somehow.
If step 2 is clean, then reconnect ADC_IN as normal, but replace the microphone with a short to ground. If the spikes are in the data, then U1 is picking up noise, either from its power supply or other interference.
If the spikes only show up when the microphone is connected, then I'd suspect that the microphone is acting as an antenna and picking up electromagnetic interference.


Possible sources of electromagnetic interference:


As you mentioned, the power supply is one potential culprit. You said you've tried decoupling, but in my experience sometimes it takes more decoupling than you might think -- either more capacitance, and/or making sure you have a high-frequency cap in parallel with a low-frequency cap, and/or minimizing the distance between the capacitor and the power pin it's decoupling.
Since your problem seems to persist for the duration of the SD card data transfer, I'd also strongly suspect the SD card interface lines. These lines could be switching rapidly and generating radiated or conducted EMI that is getting picked up by another part of the system. To remedy this, I'd recommend some series impedance -- put a 10 ohm resistor on the SCK, MOSI, and MISO lines. The resistor should be located as close to the driver as possible.


Last but not least, as @JohnRB indicates, if this problem is not digital then it will be hugely dependent on the layout, so without more information on your layout we'll all be working blindfolded here.