What shutter speed should I use for photos of people indoors?

I'm using nikon D90 with 24-70 2.8, also 70-200 2.8… What shutter speed will be safe to use handheld and not get any blur? Group shots and also people walking around casual and laughing?
To be safe, should I jack my ISO up until i'm atleast at 1/100th? Or just forget using grainy 1600 ISO and stick with sb600 flash and 100 ISO?

I'd stick with the flash and fast shutter speed. Use a diffuser/bounce to reduce strong shadows, if necessary, though beware colored walls/ceilings that could give a color cast. Digital photos use no film so try it out, if possible where you'll be taking the photos.

Step 1: Take a photo.
Step 2: Review the photo on the LCD screen of your camera.
Step 3: If the image is not blurry, take more photos. If the image is blurry, increase shutter speed.

No one can give you any better advice as we have absolutely no way of knowing the exact lighting conditions you will encounter at the event.

If you're trying to minimize blur caused by camera shake, the mininum hand-held speed is generally the reciprocal of the focal length. So if you're shooting your 24-70mm lens at 70mm, the minimum shutter speed would be 1/70th sec. In theory, the same lens shot at 24mm could be used with a shutter speed as slow as 1/24th second but, that's just theory and doesn't account for the biggest factor… You. The weight of the camera and lens, your skill-level and dexterirty, as well as the need the need to arrest/stop subject motion (among other factors) will change the minimum shutter speed required to prevent motion blur (caused by your camera or subject) from one shot to the next.

For example, if I'm shooting with a lightweight 50mm f/1.4 prime and my subject is steady and I can lean against a wall or solid piece of furniture, I can hand hold the camera at 1/15th of second and still get a reasonably sharp image. But if I've been shooting a long wedding, don't have anything to lean on and my subject is a small child, any shutter speed below 1/125th of a second with that lens is just a pipe dream. And unfortunately, sometimes I have to either bump my ISO, add a flash, or do both to get adequate shutter speeds when shooting indoors. The point lurking here is that the answer varies from one shot to the next with any given lens or camera. The basic rule remains to follow the reciprocal of the focal length as a minimum but again, other factors may require you to shoot a higher shutter speed or possibly allow you to shoot slower shutter speeds if your hands are steady enough.

F/4, iso 400, SB-600 powered at 1/8 bounced off diffuser or DIY diffuser, shutter speed at 1/125th, standing about 6 ft away. Hope that helps.

**I have no idea how the environment looks in respects to my given settings**

Depends on what focal length you are using and how much ambient light there's.

Make it easy on yourself and use flash.

1/100 is not iso. That sounds like shutter speed.