You can do a lot of research on the subject yourself. Anything on opamps by Walt Jung is a good read. Also, Douglas Self had some good information on opamp stability. Now that he's published, it looks like he closed his page.
Here are some excerpts from his book.
One thing to remember is that not all Opamps are unity-gain stable; also known as a gain of 1. Simply put, 1V input will give you 1V output or 1V/V. A gain of 2 will give 2V output with 1V input (2V/V). Lowering gain below unity is not possible and must be performed as attenuation using resistors and it will not do anything for stability. Gain of -1 is an inverted output (180°) of equal magnitude (-1V/V). Changing the gain of the circuit is not recommended unless you have (or can trace) a schematic and know exactly what to expect.
A lot of these opamps have great specs - if you're using them as instrumentation front-ends. They're
very picky and require an above average understanding of them. Many also do not have zero-offset output. Some cannot drive low impedance loads - even a 1k resistive ohm load is too much! Others don't have the correct output configuration to drive a the next circuit stage and will "rail stick" at the slightest input imbalance. Still others won't drive a modestly capacitive load without going straight into oscillation.
Then there is sensitivity to parasitic oscillations. A marginally stable opamp
for your particular configuration will ring or oscillate only under certain circuit conditions. [As an aside - this is a big reason I do not buy low-cost car amplifiers with built-in filtering. The "designer" is using the cheapest part and has not done stability analysis on the design. A simple frequency sweep that they do in magazine reviews don't pick up stability margin problems with the filter design. But, I digress - it's a topic for another day.]
Here's a good one for you. I designed an amplifier about seven years ago that used a fairly high quality, low offset op amp in a DC servo loop. I built the whole thing up and flipped it on - POP, BANG. Fuses blew. Outputs blew. After several rounds of checking, I pulled the output devices and found that my preciously chosen op-amp was oscillating, driving the output madly to both rails at around 500khz (fully clipped!). Note that I did hear a tremendous HISSS from my tweeters even though the fundamental oscillation occurred far above human hearing! I found that I had missed a key specification in the choice of the op amp - it's ability to drive a capacitve load (fairly high current) was limited. The opamp wasn't driving a capacitor, directly, but the feedback pole in the amplifier certainly is. Being a servo compensation loop, the op amp's sole job was to keep the DC operating point close to zero, automatically, reducing the requirement for a the large series capacitor that adds a pole to the amplifier's LF response. I replaced the $3 Burr Brown opamp with a 75 cent variety and it worked perfectly, holding offset to below 500uV.
I own several really good (yet expensive) books on the topic. One is Walt Jung's (pronounced 'young')
OP Amp Applications Handbook. Well worth the $80 investment (looks like it's only $51 now).
A book that was just recently published by my good friend Bob Cordell is
Designing Audio Power Amplifiers. Bob is an amplifier guy - but he's plain spoken and his book is only $38. Plus, he's approachable via email and over at DIYAudio.com