I use a Nikon D7000 with a prime 35mm/f1.8 lens. I usually shoot at the highest aperture with 200-1000 ISO, holding the camera steady and taking 6 or more shots in the hope of at least one coming out sharp.
It seems like more often than not, they don't come out as sharp as I wish.
Recently I've been thinking about when the tripod becomes significant. Is there some sort of starting shutter speed threshold to abide by?
Of course the answer is, it depends.
A common rule often mentioned is that to get sharp images hand held, you need a shutter speed that is 1/focal length used. When using this rule though you must also take your format or sensor size into account. Lucky for you, you do have a full frame(35mm) sensor so no factor is necessary. You must also consider if your lens or body has image stabilization/vibration reduction built in.
In your specific example, with a D700 and a Nikon 35mm f/1.8AF(non-VR), you should then be able to shoot hand held until about 1/35th of a second, or more likely 1/30s or so. Does this mean that every shot at 1/30th of a second will be tack sharp? No it does not. It is just a general rule of thumb that may or may not work for you and your shooting style.
If the image is critical and you have a tripod available, certainly use it with any and all shutter speeds. The technique that you have noted of shooting in burst mode and determining which is the sharpest is a fair idea and used often by today's digital shooters. But the fact is, if you care about sharpness and printing big, you will use a tripod for every shot if it is possible.
Here is an example I created with a full frame camera and a 40mm lens, shot at 1/40s over 4 shots, hand held from 3ft. Not a definitive test by any means but at least shows in practice what following the "rule" may get you:
Much more detailed information with examples of this rule in practice can be found at the following link: http://www.bobatkins.com/photography/technical/handholding_shutter_speed.html
To concisely answer your question, a tripod always matters, but practically speaking for shots slower then 1/focal length I highly recommend a tripod.
Other questions about tripods and when to use them:
A tripod always matters. With a good tripod and head, a camera will always be more stable.
There is a rule-of-thumb, which states that the shutter-speed should be faster than the reciprocal of the focal-length in 35mm-equivalent terms to get a sharp images while hand-holding.
In the case of a 35mm lens on a full-frame camera, that should be 1/35s or faster. This is just a general rule which applied reasonably well when cameras were lower resolution. Now, people often aim for twice the speed, so 1/70s or faster in the 35mm example.
There are many factors that will influence the answer to your question, and the answer can vary significantly from one photographer to the next.
The "1/equivalent focal length" rule of thumb takes this into account and recommends the point at which the average shooter should be a able to get a handheld shot with no perceivable blur at typical display sizes.¹
The practice of good camera stabilization techniques. Using good shooting posture and techniques can greatly enhance the photographer's ability to get blur free shots at lower shutter speeds. For example, many photographers who have training in marksmanship can successfully use the breathing techniques they learned with firearms when using a camera. In my younger days during the film era I could go two stops slower than the "1/focal length rule" and get photos sharp enough to display at 5X7 or 8X10. At a minimum good technique means a firm grip on the camera while holding the elbows in and bracing the camera against the face while using the optical viewfinder. On the other hand, use of the "dirty diaper grip" will almost certainly result in camera movement at even moderately fast shutter speeds.
The physical condition and abilities of the photographer. Some of us naturally have steadier hands than others. Our overall physical conditioning also plays a role in this, especially if we're shooting during times of physical exertion. Shutter speeds that work fine when we are relaxed and at rest can become problematic when we have just ran a distance to get in position for a critical shot of a live event as it unfolds. Although I've never been in the condition of a world class biathlete, who can ski cross country for several kilometers and then stop and control their heart and respiration rate in order to do precision marksmanship, I find that as I age the monopod becomes a more and more useful tool in situations that require moving faster while shooting than using a tripod allows.
The intended display size of the image. A minor amount of blur that will not be noticeable in a 4X6 print or an image re-sized for web viewing (or on the review screen on the back of your camera) will be quite noticeable if the image is displayed in high resolution at a large viewing size such as 16X20.
The effect of Optical Image Stabilization. Some camera/lens systems use compensation to counteract the effect of minor camera movement during the time the shutter is open. This is variously called Vibration Control (VC), Optical Stabilization (OS), Vibration Reduction (VR), etc. as well as Image Stabilization (IS). For each stop that an IS system is rated for, the shutter speed may be halved when compared to the "1/focal length rule". A 200mm lens on FF body with 3-stop IS could in theory be handheld down to shutter speeds of around 1/25 sec!
In general, you should only handhold your camera if the shutter speed is equal to or faster than the reciprocal of the equivalent focal length. If the image is being recorded at a size other than 36X24mm, the conversion factor should be taken into account. Based on this "rule", a 35mm lens on a full frame camera should not be handheld at speeds below 1/35 sec. A 200mm lens on a 1.5x crop body should not be used handheld below 1/300 sec.
If you are not happy with the sharpness of your photos, try using a tripod whenever your shutter speed is less than twice the reciprocal of the focal length. For your 35mm lens that would be at 1/70 sec.
There may be some other things going on here as well:
It is unclear from your question which aperture setting you are using. When you are shooting at the widest aperture of f/1.8, the Depth of Field (DoF) will be very thin. If your AF Fine Tune is even a little out of adjustment with this lens, the plane of focus may be landing behind or in front of your intended subject. Additionally, most lenses are a little soft at widest aperture. By stopping down to somewhere between f/2 or f/2.8 you might find your photos much sharper within the DoF. If, on the other hand, you are using the highest aperture of f/22, then Diffraction is affecting the sharpness of your images.
With higher ISO comes more noise. This is because the exposure values are lower and thus so is the signal-to-noise ratio. If your camera calculates an exposure of 1/30 sec. at ISO 400 and f/1.8, if you change to ISO 800 the camera will use 1/60 sec. at f/1.8. One-half as much light is competing with the same noise floor. One of the effects of the noise reduction cameras (or RAW convertors) use to minimize noise is a loss in subject detail.
¹ As with many photographic 'rules of thumb', there are also unstated assumptions involved. In the case of 1/FL, it is the assumption that the image will be viewed at a display size of 8x10 inches from a distance of 10-12 inches. If one chooses to 'pixel peep' an image at 100% (one image pixel per screen pixel), the rule must become much more stringent. Viewing a 24MP image on a 23" HD (1920x1080) monitor is the equivalent of viewing a 60x40 inch print. For a 50MP image, it is like looking at a 120x80 inch print!
If the lens you're using is the Nikkor 35mm f1.8 DX model and not the full frame model it may be the lens? I have that lens and it will not take a sharp photo until stopped down to at least f2.8 or f3.2. Kind of defeats the purpose of having a fast lens. I manage to get fairly sharp photos at f2.8 of my grand daughter and it is a $200 lens so I don't complain much, but i'm considering getting the full frame model for $500 or maybe the 50mm f/1.8 which is a full frame lens. I would try stopping it down to f/2.8, even if you have to boost the iso
To avoid camera shake 1/60 is the slowest for hand holding.
