Electronic – Cheap simple way to make an adjustable 1-10 MHz square or sine wave oscillator

What is the bandwidth of your oscilloscope? I suspect the the frequency of the oscillator may be too high for your oscilloscope to display properly.
As you probably know, a square wave is made up of many sine waves, the fundamental frequency and an infinite number of odd harmonics decreasing in amplitude which give it it's shape. So if you have a square wave at 10MHz, you will have a harmonic at 30MHz at 1/3 the level of the fundamental, one at 50MHz at 1/5 the level of the fundamental and so on. The more harmonics present the more it will look like a square wave. Any square wave can be turned into a sine wave using a low pass filter to remove all harmonics apart from the fundamental, so unless your oscilloscope has the bandwidth to display at least the 3rd and 5th harmonic you will not see something that much resembles a square wave.

Judging from the pictures I am guessing the bandwidth is around 100MHz-200MHz for your scope, the 80MHz looks like an almost perfect sinewave, the other two look to have some of the 3rd harmonic present. If your scope has ETS (equivalent time sampling) select this mode, it will display higher frequencies (providing the analogue bandwidth can handle them) but take longer to refresh. If it doesn't you will need a higher spec scope. If you give more details about your setup (scope, probe, scope input specs, sampling rate, analogue bandwidth) it will be easier to confirm one way or the other. If you are worried about breadboard capacitance (very possibly also causing problems) then maybe set it up so the signal lead is not plugged in (do something like bend it out or hang over the edge so you can get the probe to it) and try it. If you can't, then try and make sure there is no ground adjacent to the signal out.

I'm imagining this to be a low-volume reciprocating pump like those used for aerating aquariums?

If so then you may be better off just replacing the air pump with one that uses a 12V DC motor. These are commonly available, such as here http://www.robotshop.com/en/sfe-air-pump.html and http://www.ebay.com/itm/12V-DC-micro-Diaphragm-Drain-pump-Pumping-air-pump-for-aquarium-drinking-water-/171122685480?pt=LH_DefaultDomain_0&hash=item27d7b4f628.