How to check for quick oscillations without an oscilloscope
Use a RC low-pass filter and a multimeter. 

Here's how: 


Connect a 10 kΩ resistor to the point you believe is oscillating. 
Connect a 10 µF capacitor between the resistor and ground
Measure the voltage across the capacitor, let's call this voltage \$V_c\$


If \$V_c\$ is around 20-80% of the max value of the signal you believe is oscillating, then it's most likely oscillating. 

If \$V_c\$ is around +99% of the max value of the signal you believe is oscillating, then it is most likely not oscillating. 



Extra info: 

Using a low pass filter in this configuration will give you the average value of the oscillating signal. So if it is 50% high and 50% low and it is a square wave signal that is between 0 V and 5 V. Then in an ideal world you will read 2.5 V across the capacitor. 

If it is high 80% of the time then you will in an ideal world read \$0.8×5=4\$ V across the capacitor. 

If it is high 100% of the time, then it's not oscillating, and you will therefor read 5 V.