Removing spikes from multiple onboard power supplies
The RD0515 is a switching type converter. As such it is not surprising that it produces spikes in an unoptimized design. There are several things you can do to help with this situation. 


At the input side to the converter make sure to apply a very low impedance high value capacitor to supply the input current surges as the internal circuit switches on and off. It may also be necessary to place some inductor or power type ferrite bead in series with the 5V supply and the input of the converter. The added capacitor should be as close to the converter as possible and would be on the converter side of any series inductance added. 
On the outputs you may need to add a low pass filter to clean out spikes before the +/- 15V goes to other circuitry. This may need to be a multi stage filter with several series inductors/ferrite power beads with intermediate capacitors to GND.
An alternate for the outputs would be to run your analogue circuits on +/- 12V and place low current linear regulators (7812/7912 types) between the converter and the analogue circuits.


I can add the comment that the RD0515 is part of RECOM's "econoline". The parts are small, convenient and have attractive prices. But for this you do give up some on how much filtering that the manufacturer puts inside the converter. The econoline is most certainly designed to be adequate for many applications that do not care about some switching spikes. On the other hand if you select this type part then you take on the burden of cleaning things up to be suitable for your specific design. 

Also be aware that the RD0515 part does not have regulated outputs. As such the output voltages can change with load current variation and input voltage variations. If your analogue circuit designs have a dependency on the accuracy of the DC voltages and have low tolerance to power supply variations you may very well want to look at the option (3) above. 

Lastly be aware that this part should have a minimum load of 10% of its rated capacity in order to meet the data sheet performance specifications.