connection contamination soldering insulated wires

A close inspection of the joint with high magnification is likely the easiest way to get a feel for the quality of the joint. A picture would help. Flux is useful for joints like this.

30 AWG wire is my favourite choice for board-level connections such as bodge wires on PCBs or point-to-point wiring on a board when prototyping. Often it is silver-plated, and solders well. If it is oxidized then the silver is visibly blackened (happens in the UK weather!) then I cut the wire and re-strip it to reveal the fresh silver, but it needs top quality wire strippers (the "Stripmaster Lite" is excellent for 30 AWG).

For off-board wire-to-wire connections, for prototypes, if I have to use 30 AWG wire, then it is possible to crimp it to normal 0.1" pin sockets (even though the wire is so thin, it can crimp to such sockets), or it could be soldered directly to 0.1" male pin headers, by tinning the wire, and tinning the end of the pin header, and then tacking them together by just holding them against each other and re-heating.
Sometimes I'll use Blu-Tack to support one end to free up a hand. 

For a bit of strain relief, super-thin heat-shrink tubing can be used, or PolyDoh is great for covering up the joins and making the pin header into a nice connector. The PolyDoh is softened with a hot air tool and then pressed into place by hand (it's hot but doesn't burn).
For direct wire-to-wire connections without a connector, I use the same tinning and tacking method and cover with heatshrink. Usually 30 AWG wire is Teflon or Kynar coated, so the insulation is heat-resistant to the hot-air tool used for heatshrink (but it helps to have a very fine nozzle on the tool).

I use mostly 30 AWG solid core wire, so it's not recommended for off-board connections, but I do have some 30 AWG stranded wire (it costs a lot more than solid core, so I only use it when necessary; it's brilliant stuff but so pricey). The comments above regarding how to tin/tack it and using PolyDoh or heatshrink apply to solid and stranded, they both usually have similar heat-resistant outer insulation. and are usually silver-plated conductors. Most of the cheap 30 AWG wires from eBay/AliExpress etc are different; they come with softer, low-temperature, lower-quality insulation, and are probably not silver-plated either. The good 30 AWG wire is called "Kynar Wire" or "Tefzel wire". Worth buying a couple of different colors of it for on-board work, it's a great problem-solver on PCBs or proto-boards.

The photo below shows on-board connections using 30 AWG solid-core wire (blue arrows), and the off-board (orange arrow) is stranded (but not 30 AWG in this case because there was no need to use it).

Another example is in the photo below, where the white and yellow wires on the surface-mount protoboards are solid-core 30 AWG wire, but the off-board orange and green wires on the right side, are 30 AWG stranded. It could have optionally been crimped, although I didn't do that here.

30 AWG stranded crimped, for wire-to-wire connections or off-board-to-board connections is in the photo below:

A lot depends on what it is. PTFE can be difficult to strip but tends to stay the same shape while soldering the wire. Watch out for fumes though. I often use a pair of 'Helping Hands' with thin wire, but the croc clips can melt into the insulation if you get the wire too hot. One thing I do with PVC on thin wire which is not recommended is to use the soldering iron to heat the unstripped end until the insulation gets soft, possibly even turning black at the very end, and then to strip it quickly using a fingernail and an opposing thumbnail. This is much kinder to the wire and not too detrimental to the digits used. Clean the soldering iron tip as soon as possible (the stand for mine has a tray for a small damp sponge for this purpose. Don't use 'ordinary' sponge, the ones for soldering irons are specially made to withstand the heat.) Tin the wires before trying to solder them together. For ordinary enameled copper wire I scrape the insulation off with a scalpel blade or similar. There is a type of enamel that doesn't need to be stripped because it melts and acts as a flux, although it often needs to be hotter to do that.

If the wire is so thin that a bit of enthusiastic bending to and fro would cause the wire close to the soldered joint to snap, I often use a short length of heatshrink tubing to add a bit of strength. The tubing with the glue inside is well worth a try. In the absence of a heatshrink tubing heating tool (basically a hair dryer with a specially-shaped airflow guide) I use a gas (butane?) cigarette lighter to do the shrinking. Only warm it up as little as possible though, and rotate it slowly, or the nearby insulation will melt. Again, watch out for fumes. If you start to feel a bit woozy then shut the cigarette lighter down and leave the room taking the lighter with you. Leave the door open and don't go back in there for about five minutes.

The best method to join wires depends on the application and what happens to the wire in use. A small wire like this needs to not see any direct force when in use. If it must flex during use, the joints need to be carefully strain relieved. Joining a small wire to a large one often causes stress in the small one just from the weight or spring of the large one. There are lots of ways to stabilize the joint and strain relieve the small wire, but more needs to be known about the application to recommend a suitable method. One method to consider is joining the small wire to a similar small gage wire.

Here is a neat way to join wires, although I seldom use it:

https://www.youtube.com/watch?v=xBw5hrV3aZM&ab_channel=Hacktuber