As you were already told, supercapacitors are a bit (well, quite) different compared to batteries.
Batteries, as they discharge, very slowly lose voltage, going from about 1.65v when fully charged to about 1.35v when completely discharged. When the voltage gets near 1.3-1.35v, a battery can no longer provide current to a device, even though it still shows voltage when measuring it with a multimeter.
In contrast, supercapacitors can completely empty themselves, they can go from 2.75v or 5.5v (two common maximum voltage ratings for supercapacitors) down to 0 volts. Unlike batteries, the drop in voltage as they discharge is directly proportional to how much current you take out of them.
The rechargeable kind of batteries need to be charged slowly, otherwise the internal chemistry is affected, the battery overheats and gets damaged, the battery can even blow up. Supercapacitors don't have this kind of problem, they can be charged in minutes or even less, but that also means that the power source needs to be able to give them so much energy to charge fast. When they're completely discharged, supercapacitors act like short circuits so the power supply gives its maximum into the supercapacitor. As they charge up, the capacitor will start to draw less current.
AC adapters use inductors and transformers inside that buzz differently at various degrees of usage, it depends on how much current a device takes from them. Some ac adapters are unregulated, meaning they say they can output 7.5v for example, but without anything serious to use that power the adapter may output 9-10v.
Phone chargers like that Nokia are also somewhat unregulated, because it's easier and cheaper to make an adapter that's less than perfect and it's perfectly safe because the phone itself has a chip inside that wil further convert that voltage into something that can charge the battery.
Now let's get back to your circuit. Remember what I said about voltage on supercapacitors going down very fast to 0v as they're discharged?
In your circuit, you have a problem because you're using that LM317 linear regulator to produce 1.53 volts.
The LM317 regulator is a good chip but you have to remember all linear regulators work in a way that make them need a bit of voltage above the configured output to work properly. As the LM317 is a very old design, it needs about 2v above the output voltage.
So, if you have a 5.5v supercapacitor before LM317 and you charge that up to 5.5v, the LM317 will produce 1.53 volts just until the voltage on the supercapacitor reaches 3.53 volts. You're losing about half the charge the supercapacitor holds due to this weak LM317. You won't even be able to use 2.7v rated supercapacitors, because the LM317 can't output 1.53v with so little voltage.
There are other linear regulators which don't need as much voltage above the set output voltage to work, some only need 30-40mV (0.04v) and they're just as easy to use as LM317.
For example, I could suggest
* MIC2941 :
http://uk.farnell.com/micrel-semiconductor/mic2941awt/ic-reg-ldo-1-25a-adj/dp/1556737 or
* MCP1825
http://uk.farnell.com/microchip/mcp1825-adje-at/ic-ldo-adj-500ma-to-220-5/dp/1578398 (this one only supports up to 6v at input so don't use supercapacitors larger than 5.5v if you use this)
Such linear regulators will allow you to use all the charge in a supercapacitor, from 5.5v or 2.7v all the way down to about 1.6v before they can no longer output 1.53v you set, giving you longer runtime from supercapacitors.
You also need to be careful about how much voltage you put on the supercapacitors, because they have that maximum voltage rating of 2.7v or 5.5v. There are some other higher voltage ratings, or lower (2.5 and 2.3v are other common ratings), you can't just put a supercapacitor at the adapter input because you'll blow it up.
You also need a regulator to give the supercapacitors a voltage just below their maximum voltage rating. For example, if the supercapacitors are rated for 2.7v you may wish to configure the regulator to output 2.65v to the supercapacitor.
You could use a LM317 here, because the fact that there's a 2v loss on the regulator doesn't matter that much as you have the adapter producing 9-12v. A lm317 would also work to charge a 2.7v capacitor from USB , because USB has 5v and that's large enough to give the LM317 room to output 2.65v or around that value.
But if you have a 5.5v rated capacitor... it won't do. The MCP1825 would again be great to get 5-5.5v from USB and output around 5v in the supercapacitor rated for 5.5v maximum.
Remember what I said about supercapacitors acting like they're a short circuit when they're discharged? When they're empty they can draw 3-4 Amps from a regulator. LM317 is designed for 1.5-2.5 amps so it will be able to give that much to supercapacitor, but it's also important that the input of the LM317 is able give that much current.
Phone chargers, some power adapters, usb jack on your computer are not really designed to output more than around 0.5-1 amps so it's not really a good idea to let the supercapacitor suck so much current.
The easiest way to do this would be to use a linear regulator that's designed to output a lower maximum current and which will have internal current limit, not allowing the supercapacitor to exceed a certain current amount.
For example, the 1117 regulator :
LD1117 :
http://uk.farnell.com/stmicroelectronics/ld1117v/v-reg-adj-1-25-15v-1117-to-220/dp/9755829TLV1117 :
http://uk.farnell.com/texas-instruments/tlv1117ckct/ldo-2-7-15v-1-3vdo-0-8a-adj-3to220/dp/2322054LM1117 :
http://uk.farnell.com/texas-instruments/lm1117t-adj/ic-v-reg-ldo-adj-1-25-13-8v-1117/dp/9485805is just like LM317 but is designed to only output about 0.8 amps and has a current limit set around 0.9-1.2 amps so the supercapacitor won't be able to draw more than that from the input. This 1117 also needs just about 1v above the output, which means you could use it to charge a supercapacitor rated for 5.5v from usb, charging it up to about 4v.
so recap ...
[ input dc, at least supercapacitor voltage rating + voltage drop of linear regulator ] ----> [ regulator to output less than supercap rating and possibly limit current at same time ] ----> [ supercapacitor ] -----> [ regulator with as low voltage drop as possible for longer life ] -----> [1.53v]
Also keep in mind that those battery coin cell type of supercapacitors are not ideal for this type of application. You should look at the supercapacitors that look like regular electrolytic capacitors.