What's the right way to float right or left using the material-ui appbar with material-ui-next?

@Kyle You had it right :)

just add to the grid container:

justify="space-between"

With your example:

<AppBar position="static">
  <Toolbar>
    <Grid
      justify="space-between" // Add it here :)
      container 
      spacing={24}
    >
      <Grid item>
        <Typography type="title" color="inherit">
          Title
        </Typography>
      </Grid>

      <Grid item>
        <div>
          <HeartIcon />
          <Button raised color="accent">
            Login
          </Button>
        </div>
      </Grid>
    </Grid>
  </Toolbar>
</AppBar>

You need to add flex: 1 to your <Typography /> component so it pushes the <div /> to the rightmost part of the AppBar:

<AppBar position="static">
  <Toolbar>
    <Typography type="title" color="inherit" style={{ flex: 1 }}>
      Title
    </Typography>
    <div>
      <HeartIcon />
      <Button raised color="accent">
        Login
      </Button>
    </div>
  </Toolbar>
</AppBar>