"How do I measure..."
"How do i measure DC offset?"
Configure your op amp with a convenient, relatively high gain, say, 1000. Short the input. Measure the DC output voltage with a DVM. Divide by 1000 (or whatever gain you configured your op amp with). That is your input offset voltage.
"Where is DC offset mostly at on a analog or digital Circuit?"
Not clear on what your question is here. Offset voltage at the input stage of a circuit most strongly influences the total offset seen at the output because all subsequent gain stages amplify the input offset. Offset voltages arising after the input stage are amplified less and have, therefore, less influence on the output offset voltage.
"How do i measure DB gain?"
(A) If you have a voltmeter that offers dB measurements, you measure the input and output voltages with a logarithmic voltmeter and subtract (arithmetically) the input dB from the output dB reading. Since you are measuring GAIN in dB, not LEVEL in dB, the reference level of the dB meter is immaterial (e.g., dBm, dBv, dBarbitrary, will all give you the same answer for GAIN). Lots of DVMs offer relative measurements in dB these days. With these, you measure the input voltage and set that as the REFERENCE. Then the output voltage is indicated as gain in dB.
(B) If you do not have a meter that reads dB directly, you measure Vin and Vout. Gain in dB is 20[log(Vout/Vin)]
Decibels are logarithmic units of ratios. "Level" is the term referring to signal magnitudes (not gains) in decibels relative to some known reference value. Thus, dB(micro)V [I don't know how to enter a Mu for micro] is 20 times the logarithm (base ten) of the ratio of the measured voltage relative to 1 microvolt, i.e., 20[log(V/10^-6)]. Decibels are very confusing until you understand their basis, then they become very natural and convenient to use and greatly simplify many types of measurements.
The "20" in decibel calculations arises from the fact that the decibel is basically a power ratio indicator. The "Bel" (obsolescent) is the log(base 10) of a power ratio and was named after Alex. G. Bell. Since it was an inconveniently small number, requiring most values to be expressed to a couple of decimal points (since a power ratio of ten is one Bel), the decibel (1/10th of a Bel) became more commonly used. If you were to measure input and output signals in units of power (Watts), the gain would be calculated from Gain = 10[log(power out/power in)]. Since it is generally much more convenient to measure signals in volts, rather than Watts (how many of you own wattmeters?), we have to use volts squared to get values relative to power. But to square a value expressed in logarithms, you multiply by 2. Thus, decibels for ratios of signals measured in volts are multiplied by 2, i.e., we multiply the log of the ratio by 20, rather than 10.
More than you really wanted to know?
"How do i measure DB?"
See above.
"How do i measure Oscillation?"
As earlier posters have said, view the signal on an oscilloscope, being sure that the 'scope bandwidth includes the possible oscillation frequency. You can also infer that oscillation is occurring if you see an output signal level higher than you would expect from the known gain of the circuit and the input signal level (after you eliminate DC as the cause of excessive signals).
"How do i measure Hum?"
Same way you measure any signal (carefully). You want to be sure you are not including non-"hum" signals, such as audio signals, noise, DC offset, radio interference, etc. The best way is to use a spectrum analyzer so you can look for signals at "hum" frequencies (in the U.S., harmonics of 60 Hz.). If you don't have a spectrum analyzer, use a narrow bandpass filter to select and measure signals at 60 Hz, 120 Hz, etc. If you don't have a variable bandpass filter, build a set of fixed filters at harmonics of 60 Hz. See app notes from National, Analog Devices, Linear Technology, etc. for circuit details.
If you don't have a 'scope, you can differentiate between hum and the broadband noise that can get through a bandpass filter by varying the bandwidth of the filter. If the measured signal level increases by 3 dB when you double the bandwidth, you are measuring broadband noise. If the measured signal level does not change with changes in bandwidth, you are measuring hum. This is because doubling the filter bandwidth allows double the amount of broadband noise energy to pass through the filter, while doubling the filter bandwidth has no affect on the hum component. Of course, you have to know that the filter gain at the hum frequency does not change with changes in bandwidth.
Have fun.
awright