Also, what are the advantages and disadvantages for each approach?
It really depends strongly on the system, and on the character sheet design.
For D&D 3.5, I can't live without a character sheet. I can quickly fill it in with the book in hand, and be done creating a character in minutes. I know where all my combat stats are, and can find them easily. I have a PDF of a customized sheet that has quick descriptions, damage dice, casting times, etc. for all of the 3.5 spells for every core class in a compact format.
Advantages: Easy to find things. Rules and formulas laid out conveniently. Skills already listed.
For Shadowrun 4E, the character sheet is a cluttered waste of space, with ten small tables that nothing fits into, instead of one big table that just fits the things I need for this particular character. I'm better off with lined-paper. I've seen people use excel sheets with multiple tabs for the various types of gear and abilities, but they didn't really appeal to me.
Disadvantages: Too many complex, diverse abilities to cram tables for all of them into a two-page format. Wastes lots of space, and you end up scribbling over sections that you aren't using, and using the space to write in the overflow from filled sections.
For GURPS, I could go either way. GURPS has diverse abilities, but they can mostly be rolled into a few unified listings on the sheet.
When I'm playing Shadowrun over Skype, or printing sheets for a convention, I create a Word document (in OpenOffice); this is a pain to create but easy to use, and doesn't waste space.
In my experience, use character sheets so you don't forget to fill out a part of the character and a blank sheet of paper only for the session to track dynamic items like hit points, spells cast, etc. That way the character sheet doesn't get overly ruined with constant erasing and rewriting. Instead the character sheet only gets updated when new (semi-permanent) items are added to inventory or a new level is reached.
Neither. I keep everything on my iPad/phone.
- Everything's either in Evernote or Google Drive as either a straight up document or a spreadsheet (for games with lots of fiddly bits).
- Being able to show up ready to game with just a tablet (or tablet and physical dice if that's what you like) is fantastic. Carrying backpacks full of heavy crap is for teenagers/college students.
- Your GM wants access to your sheet? Just share it with him online!
- I've misplaced hundreds of pages of gaming crap over the years, I have yet to misplace Google.
- I'm already carrying it because it's got PDFs of all the books we're going to need to play and they're searchable!
Now for downsides:
- Obviously you don't want to spill coke on your expensive electronic device.
- You probably shouldn't get cheetos dust on it either. I'm trying to be less of a fat old man, so honestly not constantly eating while I play is really more advantage than disadvantage.
- It's battery will run out eventually. My iPad lasts for about 10-12 hours of gaming use off of a full charge. That's almost twice as long as my typical play session, but it's inadequate if you're going to be spending the day playing things at a convention.
- You can't crumple up your sheet and throw it at the GM when he screws over horribly. Well I guess you could, but you'd really have to work for it.
A character sheet helps keep track of the myriad rules and options you would otherwise overlook. This does depend on your game: for complex games like D&D 3.5 and Pathfinder the number of things to keep track of is huge and depends on the class you're playing, while a simpler and more emotive game may be better served with a few scribbled notes and raw roleplaying.
It's worth finding a good set of character sheets. I humbly submit my own character sheets for D&D 3.5 and Pathfinder as being quite good, but Mad Irishman's are good as well.
I like to encourage the use of plain paper over printed character sheets / power cards in D&D 4e because it forces players to understand their characters. I've found that players often pick powers that have cool names and print them out only to come to the session completely unprepared to actually use them. Players often seem to fumble with their deck of cards when their turn comes up, then say "I'll use this one" and then struggle to figure out what it does.
Even the act of writing out the text of powers goes a long way to making players internalize and understand that text.
I used keep a 3 ring binder with the character sheets inserted under the transparent flap on the front and back, and graph, lined, and blank sheets inside for sketching maps, keeping a campaign journal, and scratch paper for sundry items. However, I've recently switched over to printing power cards out of the D&D Character Builder tool and Scotch-tape "laminating" them to heavy cardstock for my "Character Sheet", and now just carry a small journal and a spiral bound set of graph paper.
I customize sheets for each game system.
I have found that, for my GMing convenience, I really want everyone using the same sheet format. This is because I use people's sheets in session, and I need to be able to find things on their character sheets.
I have also found that, in general, I prefer to include the optional data that I make use of; for Traveller, as an example, I do use homeworld data, so there is a spot for it. Likewise, I include space for both base and current attributes, and career history.
Generally, I do up the sheets in Apple's Pages or in a DTP layout program (Pagemaker or Scribus). I usually print them on 110# card.