LM2940 LDO Regulator Instability, high frequency fluctuations
Have you taken their compulsory advice re the output capacitor?

(1) COUT must be at least 22 μF to maintain stability. May be increased without bound to maintain regulation during
transients. Locate as close as possible to the regulator. This capacitor must be rated over the same operating temperature range as the regulator and 

(2) the ESR is critical -  see curve.





Added:


  C3-100uF is an SMD tant: farnell.com/datasheets/1923401.pdf With maximum ESR (@100kHz) = 0.225Ω. 


Tying down what manufacturers really mean can be a challenge.
The only capacitors with 225 milliOhm ESR in that pdf seem to be not 100 uF except a 4V one - BUT the Franell website has a number that seem to match. 

NB - in the pdf you cite they say: 


  " ... Low ESR solid tantalum chip capacitors allow delta ESR of 1.25 times the datasheet limit after mounting."


ie the ESR after soldering may have a "delta ESR"  (their term) of 125% over data sheet value - they do NOT say 125% of ds value but DELTA - implying the final value may be &lt;= 2.25 the data sheet value after soldering. Whether this is really what they intend is unknown. 

You quote the ESR at 100 kHz. The vishay datasheet shows change of  ESR with frequency. It can be 2x to 4x higher at 2 x mains frequency. Your oscillations are at 18 Mhz where the ESR is hopefully very low. Whether the 120 or 100 Hz ESR is relevant is tbd BUT as you have a 2 x mains input ripple that the regulator is dealing with this may be the instability trigger. 

As, if anything, your ESR may be too high (it seems) I'd try another cap in parallel. 

And also separately a resistor in series for belts and braces coverage :-)

Operation from a pure DC supply for testing MAY show you if mains ripple is the trigger (and may not).