Which is the safest way to get root privileges: sudo, su or login?
Security is always about making trade-offs. Just like the proverbial server which is in a safe, unplugged, at the bottom of the ocean, root would be most secure if there were no way to access it at all.
LD_PRELOAD and PATH attacks like those you describe assume that there is an attacker with access to your account already, or at least to your dotfiles. Sudo doesn't protect against that very well at all — if they have your password, after all, no need to try tricking you for later... they can just use sudo now.
It's important to consider what Sudo was designed for originally: delegation of specific commands (like those to manage printers) to "sub-administrators" (perhaps grad students in a lab) without giving away root completely. Using sudo to do everything is the most common use I see now, but it's not necessarily the problem the program was meant to solve (hence the ridiculously complicated config file syntax).
But, sudo-for-unrestricted-root does address another security problem: manageability of root passwords. At many organizations, these tend to be passed around like candy, written on whiteboards, and left the same forever. That leaves a big vulnerability, since revoking or changing access becomes a big production number. Even keeping track of what machine has what password is a challenge — let alone tracking who knows which one.
Remember that most "cyber-crime" comes from within. With the root password situation described, it's hard to track down who did what — something sudo with remote logging deals with pretty well.
On your home system, I think it's really more a matter of the convenience of not having to remember two passwords. It's probable that many people were simply setting them to be the same — or worse, setting them to be the same initially and then letting them get out of sync, leaving the root password to rot.
Using passwords at all for SSH is dangerous, since password-sniffing trojaned ssh daemons are put into place in something like 90% of the real-world system compromises I've seen. It's much better to use SSH keys, and this can be a workable system for remote root access as well.
But the problem there is now you've moved from password management to key management, and ssh keys aren't really very manageable. There's no way of restricting copies, and if someone does make a copy, they have all the attempts they want to brute-force the passphrase. You can make policy saying that keys must be stored on removable devices and only mounted when needed, but there's no way of enforcing that — and now you've introduced the possibility of a removable device getting lost or stolen.
The highest security is going to come through one-time keys or time/counter-based cryptographic tokens. These can be done in software, but tamper-resistant hardware is even better. In the open source world, there's WiKiD, YubiKey, or LinOTP, and of course there's also the proprietary heavyweight RSA SecurID. If you're in a medium-to-large organization, or even a security-conscious small one, I highly recommend looking into one of these approaches for administrative access.
It's probably overkill for home, though, where you don't really have the management hassles — as long as you follow sensible security practices.
This is a very complex question. mattdm has already covered many points.
Between su and sudo, when you consider a single user, su is a little more secure in that an attacker who has found your password can't gain root privileges immediately. But all it takes is for the attacker to find a local root hole (relatively uncommon) or install a trojan and wait for you to run su.
Sudo has advantages even over a console login when there are multiple users. For example, if a system is configured with remote tamper-proof logs, you can always find out who last ran sudo (or whose account was compromised), but you don't know who typed the root password on the console.
I suspect Ubuntu's decision was partly in the interest of simplicity (only one password to remember) and partly in the interest of security and ease of credential distribution on shared machines (business or family).
Linux doesn't have a secure attention key or other secure user interface for authentication. As far as I know even OpenBSD doesn't have any. If you're that concerned about root access, you could disable root access from a running system altogether: if you want to be root, you would need to type something at the bootloader prompt. This is obviously not suitable for all use cases. (*BSD's securelevel works like this: at a high securelevel, there are things you can't do without rebooting, such as lowering the securelevel or accessing mounted raw devices directly.)
Restricting the ways one can become root is not always a gain for security. Remember the third member of the security triad: confidentiality, integrity, availability. Locking yourself out of your system can prevent you from responding to an incident.
The designers of the secured OpenWall GNU/*/Linux distro have also expressed critical opinions on su (for becoming root) and sudo. You might be interested in reading this thread:
...unfortunately both su and sudo are subtly but fundamentally flawed.
Apart from discussing the flaws of su and other things, Solar Designer also targets one specific reason to use su:
Yes, it used to be common sysadmin wisdom to "su root" rather than login as root. Those few who, when asked, could actually come up with a valid reason for this preference would refer to the better accountability achieved with this approach. Yes, this really is a good reason in favor of this approach. But it's also the only one. ...(read more)
In their distro, they have "completely got rid of SUID root programs in the default install" (i.e., including su; and they do not use capabilities for this):
For servers, I think people need to reconsider and, in most cases, disallow invocation of su and sudo by the users. There's no added security from the old "login as non-root, then su or sudo to root" sysadmin "wisdom", as compared to logging in as non-root and as root directly (two separate sessions). On the contrary, the latter approach is the only correct one, from a security standpoint:
http://www.openwall.com/lists/owl-users/2004/10/20/6
(For accountability of multiple sysadmins, the system needs to support having multiple root-privileged accounts, like Owl does.)
(For desktops with X, this gets trickier.)
You also absolutely have to deal with...
BTW, they were to replace sulogin with msulogin to allow the setup with multiple root accounts: msulogin allows one to type in the user name also when going into the single user mode (and preserve the "accountability") (this info comes from this discussion in Russian).