- What are most people running.  Softmod's or Modchips?
I am sure most people are using softmod now with one of the many USB loaders.

- Is there any advantages [one over the other]?
Modchips work with any wii and of course require you to burn dvd's. softmods could be removed by future updates from Nintendo

- There are a huge array of hardware modchips and I looked into the Cyclone, but I notice most people talk about D2S?  Could you advise the most popular? (I don't mind using my soldering iron)
It all depends which version Wii you have. You would need to open your console and look at the chips to find out if you have a DMS, D2A, D2B, D2C etc. Most people are using softmod now though..

- If Softmod is the way to go, could you point me in the right direction?
You first need to look at your serial number and see which Wii you have. Some have different rules as to what you can do with them. Post the first characters and 3 numbers from your serial number (something like LEH104) and we can go from there. Also we will need to know which system menu you have, you can see this at the top right of the wii settings menu, something like 4.1E.

- Are updates and online play safe once modded?  (e.g. Xbox 360 requires Stealth checked games, but never 100% safe)
Yes they are safe. Currently there is nothing to detect a modded console.