Why would you plot out to 500MHz (and zoom to similar time scales) when your signal source obviously cuts off at, I can hardly tell, maybe 1MHz?
If it's not obvious to you, then listen up,
FFT shows a tail on the far left side. Maybe it's just DC offset, I have no idea. It's just a few pixels. But in any case that looks like the region of interest.
Then it drops off, pretty sharply. Like -40dB sharp. That's good attenuation for an active filter!
Since your circuit isn't being explicitly filtered, in any obvious way, that's good filtering indeed...
Then it goes down to the noise floor, at -70dBm or so. That's a pretty good noise floor. That's about a third of a nanowatt (at 50 ohms)!
Say, I wonder, what's the scope's spec sheet say the noise floor is? Never ignore and forget your instruments' capability!
Then there's a hump around 20MHz. "Harmonics" from the spiky waveform? Ambient noise (SW radio stations, SMPS interference)? Who knows. Ditto at 65.
Anyway, now you're well into the danger zone. Your proto's construction and probing isn't very good -- it's an excellent antenna, and you've got a forest from 88-108MHz. This shows up clearly on the time waveform, at 100ns/div and below. It is most certainly not "really white noise". The central peak is >40dB stronger than the background!
So what you need to do:
Shield the circuit. Ground it well, seal circuit-ground to connector-ground. Use a shielded connector. Use BNC at all, don't probe the damn thing! Because a differential probe doesn't mean anything if it's picking up a big loop's worth of RF! Especially in an urban environment with so many strong radio stations nearby!
Filter the output. Filter it to whatever bandwidth you're expecting. Dn't know how much? Well, it looks to be on the order of 1MHz, but what exactly, I don't know. Like I said earlier, it depends on how much correlation you can tolerate (which, if you don't know what you're doing... it's probably a lot).
And subsequent to the filter, ignore any frequencies that appear in the signal, well above the filter's cutoff. These are clearly unintended, not present (in the source itself), and interfering. Turn on the scope's bandwidth limiting feature (usually an option for 20MHz input bandwidth, but also options for digital filtering; beware if it varies with time/div, as is the case for "high res" and averaging modes -- always check your sanity and understand how the instrument performs the measurement, before reading numbers off it!).
Do these on your SMD board, and you should have much better waveforms.
Tim