I went to a break out session titled "Top 10 Op Amp Mistakes" at a conference not long ago, given by an Analog Devices FAE and this scenario came up. The first option with bypass caps from both power and the ground plane to the virtual ground will be better. Ideally, you'll want to have several caps each, say 10uF, 1uF, 0.1uF, even better, in progressively smaller packages. Of a 0603 and 0402 cap of the same value, the 0402 cap will have the higher SRF. The idea is to extend the SRF of the total capacitance out significantly further than any signal of interest.
You have wildly complicated your circuit, assuming your symbols mean what the schematic indicates.
First, an instrumentation amp would look like
simulate this circuit – Schematic created using CircuitLab
You show the upper amplifier, which produces OutInv, as being connected to the same ground as R3 and R9. Under these circumstances, OutInv is forced to ground (plus the input offset of the op amp), so you could profitably get rid of the op amp. Furthermore, a gain of 100 is perfectly reasonable from a single op amp, so you could do the whole thing with 1 op amp, as George Herold suggested. A simple version would be
simulate this circuit
This will have a nominal gain of 101. Since an LM358 has a maximum offset voltage of 7 mV, you could have an offset error of 0.7 volts.
Let's assume, though, that the two input grounds are instead a separate ground, isolated from the output ground by some common-mode voltage. In this case, you have connected the upper op amp incorrectly. R1 should be connected to the - input of both amps. However, if you do this, you'll have gain of about 220, rather than 100. The equation for gain for an instrumentation amplifier (assuming R2 equals R3, R4 equals R5, and R6 = R7) is$$G=(1+\frac{2R_2}{R_1})\frac{R_6}{R_5}$$ Alternatively, depending on the output impedance of V1, you could simply use a single op amp set up as a difference amplifier with a gain of 100. You've shown the V1 as a voltage source, so this seems perfectly reasonable.
If you're worried about a stable output, I assume you're worried about noise from the input. This is best handled with an RC filter between V1 and the input.
Best Answer
You need to have a negative supply voltage on the op-amp to get a negative voltage out of it. A rail-to-rail op-amp can, at best, approach the supply rails at the output. A non-rail-to-rail type might only get to +1.5V if the negative rail is 0V.
So, the supply voltage must exceed the voltage you need out of the amplifier. If you need +/-4.9V, a +/-5V supply may do with a rail-to-rail output amplifier. If you have an LM324 it might only get to +3.5/-4.5V with a load resistor to ground.