Sometimes these user interfaces can be a bear to figure things out. Google did a rather pesky thing of burying the control for this deep in the settings option tree.
1. Go to "Settings".
2. Scroll down to the bottom and click "Show Advanced Settings"
3. In Advanced Settings, scroll down to "Languages", then click the "Languages and Spell Checker Settings"
4. In the dialog that pops up, you should see at the bottom an "Enable spell checking" checkbox. Be sure it is checked. Also, be sure your language of choice is listed and the status is set for active spell checking (if not, then click "Add").
5. You may want to "restart" Chrome just to be sure the settings are in effect.
Now, if you've got it checked already and Chrome still isn't offering you any spell checking services, then your installation has been compromised somehow. I do have one solution that may help. If you don't have many extensions installed in Chrome, then do the following:
1. Back up your bookmarks (go to Bookmark Manager, --> Organize --> Export)
2. Uninstall Google Chrome.
3. Reinstall Google Chrome, using the latest from Google.
4. Import your bookmarks.
5. Reinstall any extensions you use in Chrome.
You should be back to "as new" status and I'd expect you will see your spell checker option working. If not... well, that's very bizarre and I'd go visit the Google Chrome forums to seek their help.
As for IE? Well, I think it is managed via 3rd party Add-ons. Check out
How To Geek for a sample set of instructions.